I'll try to answer your question, sorry if it doesn't satisfy you

Who would these troops be

So before getting into mobilizing the troops, it's better to know what kind of troops that will join his army first. You've listed some, like the Kapikulu, Timarli, and the Eyalet soldiers (Actually, the Timarli unit is still under the category of Eyalet troops). Sekbans as you probably imagined (musket-bearing infantry) only started to show up during the late 16th to early 17th century. Before that, Sekban was actually a regiment in the Janissary Corps.

First the Kapikulu. They were the Sultan's personal standing army, Comprised of the Janissaries, Kapikulu Sipahi (Household Cavalry), The Artillery Corps, and the Technical Branches. The janissaries were the nucleus infantry force of the Ottoman army, with their volley and discipline along with coordination with Sipahi cavalry being the deciding factor in much of Ottoman battles on this period. The kapikulu sipahi were at first a royal guard unit solely to protect the sultan, but over time this cavalry unit became larger and transformed into an elite unit rather than royal guard. With their increasing strength, they often take part in the field on final moments to reinforce an attack or turn the tide during crises. The artillery corps were one of the oldest and most prestigious branch in the standing army, their fortress artillery was important in sieges and their mobile artillery was as important, being positioned in the centre of the battle formation to provide supporting fire. The technical branches under the artillery corps were comprised of the Cebeci Ocagi (armourers), Top Arabacilari Ocagi (wagoners), Humbaraci Ocagi (bombardiers), and the Lagimci Ocagi (miners). The Cebecis were responsible to manufacture and repair weapons and equipments of the kapikulu troops. The Arabacis were responsible to provide wagons for the Tabur formation, and sometimes to transport equipments too. Starting in late 16th century, this branch has lost its prestige and their role was reduced to only as transport units. The Humbaraci were bombardier artillery units equipped with explosive mortar shells, but they only played a minor role and is mostly marginalized until their reorganization in the 18th century. Lastly, the Lagimci were engineer corps responsible to mine enemy walls, perform countermining operations, and build earthworks. They often aided by civilian auxiliary units in their engineering tasks and sometimes they also tasked to build and maintain roads and bridges.

And second, the Eyalet (provincial) troops. Primarily comprised of Timarli Sipahis and their troops, the Serhadkulu (Frontier Troops), and various auxiliary units. The Timarli Sipahi make up the bulk of Ottoman cavalry, and performed like a conventional light cavalry. They were granted lands in exchange for military service, where they must provide their own equipments and brought armored retainers with them for each 1,000 to 5,000 akçe. The Serhadkulu units were comprised of various units, from the Akincis, Delis, Gönüllüs and Crimean Tatars. They were supplied by governors of the frontier provinces like Bosnia and Semendire (Smeredevo) and mainly functioned as raiders to enemy territory and occasionally support the army. Lastly, the Auxiliary were also comprised of several units that performed various duties. For example the Azabs were cheap expendable units to bear the brunt of enemy attack, the Derbendcis that performed guard duties, Martolos who were initially Christian fortress guards but later became an umbrella term for Christian soldiers, renegades, and agents, and the yayas and musellems who were essentially light infantry and cavalry forces. However, starting at the middle of the 16th century these auxiliary forces were starting to less being a soldier but rather as labor battalions performing menial duties as to curb their numbers and tax-exempt status in order to increase tax revenues, and most of these auxiliary forces were abolished at the late 16th century

Now about the Sekbans, they were initially a Janissary palace guard company tasked to guard Istanbul when the bulk of the corps was absent on campaign and to reinforce Janissary loyalty and discipline. However, later on the late 16th century, with the increasing demand for musket-bearing infantry, the government starting to recruit mercenaries and irregulars to match their rival's firepower. These irregulars were labeled under various names, like Levend, Sarica, Tüfenkendaz, and Sekban. Though i don't know what makes them to be named similar to that certain Janissary units, as Sekban literally means "Houndkeepers"

How would his government gather troops for this campaign

For the kapikulu soldiers, A roll call would be held on cities where the kapikulu troops were stationed, and those kapikulu must mobilize under their leaders and officers (For example the Bölükbasi for the Janissaries and the Bölük agalari for the kapikulu sipahi) and march to the provided central gathering points to join the main imperial army, usually Istanbul but sometimes Belgrade and Diyarbekir also designated as gathering point depending on the campaign. Of course, sometimes the mobilization effort encountered difficulties with persuading troops to actually rise up and join the army. Various methods were used to encourage mobilization, from threatened punishments like removal from payroll to offered incentives like wage raise. The same also goes for the Serhadkulu units, where they were mostly a part of a governor's retinue, and said governor will be notified to mobilize his retinues alongside the kapikulu soldiers. For the timarli sipahis, the Alaybeli (colonel), leader of each timarli sipahi regiments will notify sipahis under his command to prepare three to four months before the campaign season and when the time came he'll make sure all his sipahis were present. Afterwards, the alay then gathered in the province center with other alays and moved to the regional assembly area under the command of Sancakbeyi (province governor). Afterwards they will be inspected by the Beylerbeyi, to ensure that the basic standards of weapons and equipment were met, then the units marched to the provided central gathering units to join the main imperial army. The auxiliary one is rather complicated and admittedly i don't know much about them. The Azabs for example, were first volunteers recruited for one particular campaign, but as time goes on they evolved into semi-mercenaries that offered their service as part of a governor's retinue, so they also must be mobilized alongside the Serhadkulu units. Other auxiliaries like the Derbend, Voynuk, Pandour, Eflak, etc usually performed guard duties and thus doesn't join the main army. Auxiliary labour battalions to my knowledge were only mobilized as the need arises.

End of Part 1

the approximate balance between kapikulu, sekban, timarli, and eyalet soldiers

In the early to mid 16th century, generally the timarli soldiers outnumbered their kapikulu counterpart. Halil Inalcik noted that on 1528, there are around 24,146 kapikulu soldiers (including Janissaries, Sipahis, Cannoners, etc) while the kilic timarli forces numbered around 37,741 men excluding their cebelu retainers, which could add 60,000 more men. Similarly, Rhoads Murphey calculated that the potential timariot force (including retainers) on 1527 to be 90,000 men (but it's considered that in real terms the total of timarlis in the field rarely exceed 73,000) while the kapikulu forces only have a grand total of 18,689 men. Of course this will change, as Hungarian and Austrian firepower were steadily increasing and siege warfare became even more important, the Porte expanded the kapikulu corps, especially the janissaries to suit the new battlefield. In 1574 for example, the kapikulu forces rose in number to around 29,175 men, and later in 1597 it rose to a whooping 58,527 men, most of them (21,094 on 1574, 35,000 on 1597) were janissaries. In 1528, Inalcik calculated that there were around 30,180 auxiliary soldiers, the majority of which were the combined auxiliary units of Anatolia and Rumelia comprised of Yayas, Musellems, and various nomad forces numbering around 15,180 men. Following them were the registered akinci raiders which numbered around 12,000 men, but their number could swell up to 50,000 men during campaign. At 1582, the numbers of these auxiliary units were reduced to mere 8,100 men. Likewise, the akinci raiders also starting to lose importance and is replaced by the Crimean Tatar raiders which numbered around 50,000 men during the late 16th century.

To kill two birds with one stone, i think you already have an idea on the composition of Ottoman army in the 16th century, which were mostly cavalry-based on the early 16th century, but on the late 16th century they started to become more infantry-based. Now on artillery, we can estimate their number from the numbers of cannons cast by the Imperial Cannon Foundry. For example, between 1522 to 1526 there were 1,029 cannons cast and between 1527 and 1528 there were 148 cannons cast, with the majority of them being small to medium sized guns like darbzen, sakaloz, and prangi. The Ottomans often brought hundreds of them during battles and sieges, for example during the siege of Szigetvar on 1566, there were around 280 darbzens used to besiege the city.

how expensive this would all be

Excluding logistical costs and whatnot on campaign, on 1528 the salaries for the entire kapikulu soldiers were 65.88 million akces yearly. As their size grows, their salary become higher, with 76.65 million akces on 1547 and 127.316 million akces on 1567. On 1528, It was estimated that the revenues goes to maintain the timarli sipahis were amounted to 200 million akces. So on 1528 the total expenditure for the army was around 265 to 285 million akces, with the entire state revenue of about 537 million akces. Unfortunately i couldn't find a more detailed data for later years. The auxiliary corps doesn't get paid, but many of those men were exempted from extraordinary taxes and they also paid their tax on lower rates. On 1528, the tax exemptions for the yaya and musellems amounted to 3.08 million akces yearly and on 1534 2.65 million akces respectively.

Well that's about it. If you have any additional questions i'll answer if i can

Sources:

Ottoman Warfare, 1500-1700 by Rhoads Murphey

A Military History of the Ottomans by Mesut Uyar

An Economic and Social History of the Ottoman Empire Vol. 1 by Halil Inalcik

Guns For The Sultan by Gabor Agoston

The Ottoman Empire, 1300-1650 by Colin Imber