That seem...unlikely that Slovakia part fall totally in Mongol hands. Too close from Bohemia, and if I remember well, more fortified than neighbours.
In fact, I don't see a conquest or vassalisation really doable in this situation. Having a tributary situation Russia-like on the other hand is easily doable.
That the central or eastern part of Hungary pay tribute is possible, tough, but I don't see this situation last when someone rise again in the pannonian plain (turning the whole thing as an hungarian Moscovy) or if Transylvanian manage to cut the roads.
For the consequences, critically if Bohemia tries to claim back part of Hungary, it could turn not really uneasy for them. They didn't have to be afraid of a Mongol control, and could make the junction up to Austria.