I think that a Mongol settlement in Hungary would result in a Muscovy-esque state emerging in either or both of Germany and Lombardy. Whichever city defects first to the Mongols (Venice, perhaps?) will get tax privileges over the rest, then have the opportunity to start expanding once the Mongols disintegrate...
Depending on what happens there could be a Khanate in Constantinople as well, if they can get there early enough the Greeks would probably let them in.
It also isn't a Mongol wank without a Mongol Egypt.

And if they can get Egypt, then perhaps they can pull an Ummayad, cross the Maghreb and then straight up into Andalusia, which is at this point in time a collection of petty kingdoms in the south....