What happened to Poland and Austria-Bohemia? What happened to the Hungarians, Slovaks, Romanians, etc in Hungary?
Thing is Mongol-Hungary is simply too easy of a prey for whoever is around it (Rudolph of Habsburg already had his own designs for it) and the Carpathian basin is too remote from the Mongol powerbase.
But let's handwave this, the basic: no one will like the mongols there, Whoever Khan that rules it will have to play smart and not let himself get crusaded, have some Hungarian lord submit to the hordes and assimilate, those would help the khan in fitting himself in the European setting, since I don't see this... Cumongaria having the bulk of Mongol military it will need an Alliance, probably with the Habsburgs, to survive for a while, which means almost immediate conversion.
For effects in Europe as whole... too many, the Arpads intermarried a lot with Poland and the balkan principalities, also let's not forget the Angevin golden era in the 14th century, this new realm will create a large vacuum, as it will need time to establish itself. I know absolutely nothing about the Balkans at the time so I'll abstain.
Hope I helped with some insight, my knowledge of the Mongols operations in Europe is kinda... lacking.