POD: 1241 goes even worse for Hungary. The Mongols reduce the remaining fortifications, calling in reinforcements from the Home Steppe & various auxiliaries to garrison those left intact. Béla IV's head is delivered to Subutai's camp near Budapest by the beginning of 1242, while Ogodei hangs on for another few years.
The Mongols remain the supreme rulers (they send reps to the kurultai, but most hang around), the surviving Maygar nobles curry favor by learning their language, and various churchmen start working on a Latin-based orthography for evangelization purposes.
What are the odds of a Mongol speaking state remaining part of Western Christiandom? And how far would it diverge from the language of the 'home' Mongol lands?