So I thought of how I should answer this. Unfortunately a lot of it is not answerable I don't believe. For the daily routines of daimyō, we have that of Date Masamune, as recorded by his cousin. However, it is but one example, and we don't know if the recorded routine is Sengoku or the early Edo. And of course, surely the routines of daimyō greatly differed from one another.

So rather than dig out Masamune's schedule, I'd like to talk what daimyōs did as politicians and administrators, rather than as generals.

Without a doubt, the final goal of the Sengoku daimyō was to be able to control as much territory, and to be able to mobilize as much resource as possible. The way to do this was through political control. But what that control looked like actually differed greatly from place to place, due to geographic and pre-existing political considerations.

For instance, in the early and middle Sengoku, the Kantō plain was fought over by the Uesugi, as Kantō Kanrei (Deputy of Kantō), and a branch of the Ashikaga family who was placed in charge of Kamakura, but was ousted from the city prior to the Sengoku, and took up position in Koga and so was known as the Koga Kubō (Ashikaga Lord of Koga). Both sides derived their legitimacy to command the other families of the plain from their position within the hierarchy of the Muromachi Bakufu. As such their way of control of the Kantō was wherever possible, to try to restore "normalcy". Where they can, they promised to restore the lands and incomes of aristocrats in Kyōto, the temple lands, and even lands of warriors, many of whom had acquired distant holdings during peace time that were disrupted by war. They issued laws and orders for traditional rights to be respected, and for realms to not be violated. Even when they couldn't realistically restore lands due to political reality on the ground (the land requested was not under their control, or had already been awarded to someone else for loyal service), they often offered compensation, or at least promised to either restore the land or to offer compensation. In the long run, this proved to be their downfall, for such system meant warriors were quick to switch to whoever offered the best deals and vassals with their own power bases could even rebel against their lords, which prevented both the Kantō Kanrei and Koga Kubō from forming consolidated, solid base of power. But in the short term, that was what they had to do, for that's where their legitimacy laid. The Uesugi Kantō Kanrei in particular, though they had castles, opted instead to hold court in "camps". In theory, the camps were movable. In practice, the camps were used years, even over a decade at a time. From there, they co-ordinated the wars they were fighting, and conducted political/religious rituals. Their followers also spent the majority of their time at the Uesugi camp, though coming and going depending on the situation and their loyalty.

In contrast, clans like the Takeda, Imagawa, Oda, and the Hōjō who would eventually defeat the Kantō Kanrei and Koga Kubō, derived their power from a single base of power backed by the clan's direct military strength. As such, their method of control was to build a central castle, and order their vassals to come live in the surrounding town. Branch families were given powerful positions or placed in key positions or strategically important castles. Rival clans were often forced to adopt a son of the conquering lord, to neutralize it and place a family member in a key geographic location. These lords would issue a slightly different set of laws, ones that were new directives about what to and to not do in the lord's realm, how taxes were collected, and land and trade dispute settled. While many had precedence in the Muromachi Bakufu such as the kenka ryōseibai (to punish both offending parties of illegal fights for disturbing the peace), others, like rakuichi rakuza (cancelling the power of guilds and cutting taxes at select markets) of the Rokkaku, Oda, and Imagawa probably do not. Importantly though, these law codes were implicitly understood to be backed by the power of the daimyō, and not the Muromachi Bakufu. The Imagawa in fact made this explicit, staying in the second revision of the law code that the law was backed by the power and control of the Imagawa lord and not the Bakufu, and therefore any old privilege would only be recognize if the clan in question follows the orders of the Imagawa lord.

Important side note here: what determined which path the daimyō took was not just its ties to the Muromachi Bakufu. For instance, the Imagawa was a branch of the Ashikaga that could produce an heir to the Shōgun should the need arise, but went with consolidation. Meanwhile the upstart Mōri of Aki and the not-quite-to-the-same-extent upstart of the Nagao (later Uesugi) of Echigo went with trying to act as overlord of a coalition.

Daimyōs also worked hard to improve the economy of the realm, for war is essentially throwing money and manpower at the problem. As mentioned above, legislation was one way to do this. They also ordered the development of mines, put special policies or people into trade (especially the new seaborne trade), and ordered the construction of dykes, irrigation projects, and the opening of new farms.

Connection with the central court at Kyōto was another way daimyōs boosted their legitimacy and solidified control. Apparently no one paid attention to the contradictory situation of issuing laws independent of Kyōto. Daimyō often paid large sums to the emperor's court, as request to this ceremony or restore than palace. In return, the court conveyed on daimyōs official ranks and titles. Sometimes daimyōs outright petitioned (in effect bought) the ranks and titles. An officially sanctioned rank and/or title was highly prized, but even unofficial ones were covetted. Daimyōs got into the habit of assigning ranks and titles to their followers as rewards, and we start finding, for example, people in Shinano being named "Lord of Awa" despite never having set foot in Awa. Some lower samurai also just went ahead and declared themselves holder of these titles, but that's another topic. Marrying women of aristocratic families were also popular. As were inviting people of the arts, poets painters kick-ball specialists, often aristocrats or monks, to come to the court of the daimyō, where prestigious festivities would be held, sometimes a day, sometimes a few days, sometimes even weeks of just sitting around and doing poetry with other vassals and the famous artists. These cultural activities were not regarded as a waste, but were considered very prized by the samurai, indeed by all section of society (for instance a famous tea cup could be worth a castle or two). As such they not only increased the daimyō's prestige, but also worked to tie his vassals to himself, for it was the daimyō who offered access to such high culture. There was a diplomatic side to this as well. The status of the Shōgun and aristocrats meant that their words held weight, and so daimyōs often ask them to intercede as a neutral third party in order to bargain a truce or a peace deal. This of course meant that the Shōgun and court could still issue orders that, while not always followed was never ignored. For instance Oda Nobunaga was ordered by the court to return lands of the court that he accidentally seized, and as late as 1576 daimyōs still heeded the call of the Ashikaga Shōgun to mobilize troops against the Shōgun's enemies.

On the diplomatic front, alliances were essentially achieved by marriage, whether it was the daimyō himself marrying or his heir(s) marrying, it was marriage ties that solidified an alliance, for the wife was also the hostage her husband's clan from her birth clan. It was no doubt hoped that their offspring, with blood from both clan, would be able to continue the friendship between the two. But very often, this did not work out. Interestingly other forms of hostages were used too when marriage was not an option. Hōjō Ujiyasu allowed Uesugi Kenshin to adopt one of his sons to solidify a (in the end very temporary) alliance against the Takeda, while Matsudaira Hirotada gave his son and heir as hostage to the Imagawa to solidify their support against the Oda. Marriage with a vassal family was also used to build loyalty, and as mentioned above, so was adoption.

Finally, one of the most important activity of the daimyō was to issue the andōjō, or certificate of guarantee of land rights, to vassals large and small. Though the Japanese did use precious-metal currency, their value was measured and backed by agricultural produce (of rice), which means land. As such, land rights were very important to all samurai, especially in times when disputes often ended up being settled with violence. This is why so much law dealt with land ownership and rights, and the dealing with land disputes. Presumably most of this would be solved by other bureaucrats, but the daimyō did sometimes have to get involved between land disputes of more powerful vassals, just like they had to get involved with succession disputes (often solved by having the vassal adopt one of the daimyō's sons or relative as mentioned, but also often not). The samurai's obsession with land also meant daimyō worked hard to use land rights to win over independent or enemy low(er)-ranking samurai. As mentioned above, this was done by issuing land guarantee of the samurai's current and/or ancestral holdings that may-or-may-not be in his hand, or to promise new lands from future conquest. This was actually extremely effective, and used by daimyōs all over Japan no matter their ruling and management style.

What made a daimyō a good politician then, was how well he could carry out the many complicated tasks described above. This might not be the exact answer you were looking for, but I hope it answers your question.