I would like to add that I have been reading the book What if in History and in it one of it is about Mongol invasion of Europe by Cecilia Hollande. She basically states that the mongols could use Hungary as an effective base due to the grassland and plains found their. Basically by the defeat of the Hungarians Europe had no real army to speak of to oppose the mongols. As said above.
Likewise the german states hated each other and tried destroying each other. Most likely to circumvent the castle problem Subodei would exploit this volatile situation in Germany by using divide and conquer siding with one lord against another lord and then taking over this lords lands. Except he would do it on a wide scale and im pretty sure many of the German lords at least near the polish-hungarian borders would immediately surrender to the mongols if they truly invaded.
Otherwise with the large enough force as assembled by mongols split into the four armies cimon mentioned Poland would be mopped.
If no pasteur were to be found here is the scary part the mongols would "convert" the area into pasteurs for their horses like they did in Northern CHina. This basically means the forests would be most likely burned down the cities completely razed thus making it possible for the mongols to Pasteurize the area if they really needed to which is
.
You must understand also how mongol intelligence were experts as such according to holande they would strike after invading from the danube into vienna the low countries overuning Antwerp, Ghent, Brughes to seek Pasteur for thier horses. They would do this by going through northern German(Actual Plans) through Liegnitz and onto Antwerp and so forth. Meanwhile another army would advance from Vienna. WIth the loss of the low countries the financial center of Europe would be wiped out in effect. Likewise the rhine delta would become a swamp if the mongols depopulated it most likely would happen.
As such the pasteur problem pointed out really does not hold true because if the mongols really wanted to they could force pasteurize parts of europe to feed their horses by Pasteurizing it.
Cimom otherwise explained the stuff well.
Now holding all this ASB. Basically once Ogedei dies and maybe batu the civil wars will make it impossible for the mongols to control this territory leading to a massive loss of it. It ould also lead maybe to a nestorian christian Khanate centered in Hungary if in order to reconcile with Guyuk Batu gave him the lands in Hungary and Poland. . But that is it. THe most the mongols could hold would be poland, Hungary, and maybe croatia but they could successfully invade Europe up to Paris holding it simply not possible due to the massive populations that inhabit these areas but you can expect massive depopulation in parts of europe especially Germany due to its forests and cities. Plus about castle warfare mongols need not take castles why if they sieze the farmland and the cities the castles surrounded will have no way to get supplies and surrender on thier own.
Edit nevermind I think Guyuk would have become Great Khanso someone else might rule that area as a nestorian christian khan.