The assumption that 'Mongols lacked siege skills' is a large oversimplification. In the period mentioned in the original post, i.e. 1240-1258, Mongol Empire was the biggest empire in the world (second largest empire in history) spanning from Yellow Sea to Black Sea and from Southern Siberia to Indian Ocean, meaning that Mongols ruled over many different peoples and were able to use their knowledge, technology and organization to further their own goals.

To address the original question, the siege capability of the Mongol armies was extremely high, mainly thanks to the knowledge and technology of the Han Empire. Accounts of both the Siege of Baghdad and the Battle of Mohi mention the presence of Chinese engineers performing both the essential engineering tasks, as well as constructing numerous siege machines. During the conquest of Rus that began in 1236 under formal Batu-khan's command (in reality, it was Subutai that has overseen larger part of the campaign) Mongols were able to successfully besiege and capture fourteen at least thirteen cities, including Ryazan, Kolomna, Moscow, Tver, Torzhok, Rostov, Volodimir, Pereyeslav, Kozielsk, Suzdal, Chernikhov, Glukhov and finally Kiev, fall of which marked the end of Kievan Rus. The duration of sieges differed vastly, and while Ryazan has been captured after six days of constant assaults and Torzhok fell after eight days of siege, Kozielsk managed to defend for two months, until Batu's forces were aided by reinforcements led by Kadan and Buri, what is usually explained by Batu's reluctance for an all-out assault due to the smaller number of forces under his command. Mongol victories that allowed them to capture or raze Ruthenian cities are a good testament to the siege abilities of Mongol soldier and commanders. Engineering acumen of the Subutai army is also confirmed by their actions at the Mohi plain, where Mongols were engaging fortified Hungarians while the smaller detachments built a temporary bridge (possibly a pontoon one), allowing attackers to cross the river Sajo and outflank the Hungarian forces.

Now, one may wonder why the Mongols did not use the same tactics in their three campaigns in Poland in the years 1240-1288. It is generally accepted that these incursions were not the all-out wars aimed at the conquest of the territory, but rather small-scale operations to support other, more important goals. The first incursion in 1240 was an attempt to engage Polish forces and prevent them from supporting the army of the Hungarian king Bela IV, with the Hungary being the real target of the campaign. The second attack, led by Burundai in 1259 was also a relatively a relatively small-scale operation meant to harass and plunder the countryside and to prevent the formation of the anti-Mongol alliance between Polish and Ruthenian rulers. It should also be noted that in this campaign the cities of Sandomierz and Krakow were besieged, with the latter being successfully overrun, although the defendants of inner keep managed to defend. The third invasion that begun in 1287 was conducted with rather limited forces, with the estimated strength of the Mongol forces being 30.000 people under Telebuga and Nogai. At the same time, Leszek the Black, Prince of Cracow alone could have mustered roughly 20.000 people including 5-6 thousands of cavalry (although Mongols probably did not know that Leszek's relations with other Polish prices were strained at best and thus the chances to receive reinforcements weren't too high) what made large sieges risky at best, as the besieging army could have been easily ambushed by substantial Polish forces. Attempted sieges of Cracow and Stary Sącz were not successful thanks to the well-prepared defense and the alliance between Leszek and Hungarian king Laszlo IV that allowed Polish-Hungarian forces to ambush the Mongols, dealing heavy losses. Seeing the futility of their attempts, Telebuga and Nogai withdrew from the territory of Poland.

The aforementioned events are a good illustration that the Mongols were more than able to successfully siege the cities in Northern Europe, so their successes in the Middle East crowned by the Siege of Baghdad in 1258 are by no means an exception.