I've always thought the Magyars were the best example of how an established Central European Mongol state would actually function - a permanent base in the grasslands of the Alföld, from which regular raids would be launched on the rest of Europe. To manage that you need a total disintegration of 13th century Hungary, but this was probably doable given time - the Mongols were able enough at tearing down the walls of Esztergom, and reducing fortresses across the rest of the country was probably not beyond their capabilities. The key thing is to do it immediately; when the Mongols withdrew, the Hungarians got nearly 50 years to prepare for the next attack, and fortified their country very effectively. If the Mongols don't land the knockout blow against Hungary in the 1240s it's unlikely they're going to get another chance.
If Batu keeps the center of gravity of his realm at the Volga, the Alföld is probably going to some Jochid viceroy, or maybe he sets up his son as his deputy there. Lombardy is likely going to be their favorite stomping ground, just as it seems to have been for the Magyars, but the Magyars were able to raid as far as Spain and the Mongols were hardly less enterprising.
Lombardy was attractive to the Magyars because of its wealth, congenial terrain, and political disorder, and it's not much different in the late 1240s when the attempt by Frederick II to control Italy finally collapses. The HRE is fraught with civil war not long thereafter and the way is basically open for the Mongols to do what they will with northern Italy.
(It's often mentioned that Sartaq, Batu's son succeeding in 1255, was a Christian. It's a ridiculous flight of fancy, but I've always liked the idea of Sartaq not being assassinated, losing a civil war for control of the Russian steppe to his Muslim uncle Berke after Batu's death, fleeing to the Alföld with his supporters, and eventually being invited into Italy by the Pope to protect him against Manfred of Sicily. Then you get the weirdest state of all, a nominally Christian Khanate of Hungary-Italy. And since the imperial throne is vacant and a grateful pope is all you really need for the crown, why not have Sartaq get it so you can end up with a full-blooded Mongol Holy Roman Emperor.

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