European kings stuck playing whack a mole because they could not exert amount of terror that mongols could. After they kill every living thing in a few cities, by every living thing I mean dogs and cats included. The resistance will subside. If they are smart they will then order all castles on their territory(except few) to be destroyed by owners, those that are not will suffer fate everyone knows they can inflict. Terror is a weapon and Mongols were masters of it.
Europe will get decastlefied and quite quickly.
Of course, terror works. However, at the siege of Esztergom, after the mongols captured the city, they garrotted all the women in front of the castle garrison, and they held.
For the record, the fact, that the garrison was mainly spaniard and italian, probably helped to make that decision a bit.
As for decastleficitation, the mongols tried something like that, with the seal of the king (lost at Muhi) they "forged" orders in the kings name ordering the nobles to accept the mongols as overlords, turn over the castles and co. As far as we know, not a single noble obeyed.
Maybe the terror backfired, maybe it was the usual nobleman stubborness.
Another thing: they cannot recruit - or more like, press gang - new troops in the new territories. Not in Hungary - despite, they were pretty much in control much of it - not in the kievian parts, not in the russian principlaties.
Wich i find interesting (did they not tried it? or tried but failed?)
But IMHO the most important part: they (maybe Ögedej) realized, that any further offensive movement or simply just holding their gains not worth it. Needs more (maybe much more) troops, time, and there is a good chance, that at the end of it, they will stuck into a swamp of castles, armies, unfavorable terrain far, far away from home.
And for what?
(And IMHO again, later the ottomans just happened to stuck into the very same swamp.)