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?
Hm... let's see what we can do without too much help from ASB:
So Ogodei hangs on for another few years and the Mongols conquered Hungary, Poland, Germany and the Balkans; or parts of these countries - it does not matter here, the main point is they keep Hungary as a part of their realm under strict control.
Let us be generous and after the death of Ogodei we have a new Universal Great Khan in Karakorum who loves Batu-Khan and the House of Jouchi in general. (For obvious reasons chances of Batu-Khan to be elected are zero.)
But I am afraid that the continuation of the Great Western Imperial invasion into Europe is out of the question. The complete conquest of the whole Europe would have been possible if Ogedei had lived 6-10 years more.
So best case scenario for Batu-Khan is to get a lot of ethnic Mongols in his disposal.
We cannot expect from a new Universal Great Khan to be dumb as a cork and give 40 000 Mongol-speaking households to the Jouchid Ulus / the Golden Horde as it was de facto the first independent part of the Empire and with these Mongols Batu-Khan would be stronger than the Universal Khan himself.
But let us presume that our new Great Khan is moderately stupid and he gives Batu-Khan 10 000 ethnic Mongols. Oh, what the hell! The Great Khan is so generous (and drunk) that he gives Batu-Khan 20 000 Mongol households to me removed from Mongolia proper into the Jouchid Ulus (the Golden Horde) for a permanent basis.
Even given 20 000 households speaking the Mongol language who are the best fighters in the world it is still out of the question to continue the conquest of Europe. And this is not because that is not enough considering immense power base among nomads of Kipchak Steppe; the main reason is that centrifugal forces in the Mongol World Empire are in full swing already and the main concern of the Golden Horde is watching it's back - against other Mongols.
Now imagine yourself to be the Khan of the Jouchi Ulus (Golden Horde) which stretches from Hungary in the West to the Central Asia (to the border with Afghanistan). If you suffer from severe case of European exceptionalism you would place these 20 000 Mongol households in Hungary - into the most Western part of your realm.
But the most reasonable location of your best fighters is settling them in the geographical centre of your Ulus - somewhere between Volga and the Crimea.
Actually these ethnic Mongols would have been spread as a thin layer all over the Golden Horde, especially in the Great Eurasian Steppe among Turkic-speaking nomads.
And the general pattern is - everywhere where the ethnic Mongols were a (ruling) minority over a (subject) majority of the Turcik-speaking peoples - in a few generations the Mongols forget their Mongolian language and become properly Turkicised.
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So, the odds of a Mongol speaking state remaining part of Western Christiandom are close to zero. (With your POD of course.)
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