Europe was too fragmented to be conquered by the Mongols. Their way would have been blocked by castles and town walls, each originally thrown up by one group in conflict with others—by a fractious nobility against its kings and against its own peasants, and by townsmen against them all. Above all, “there was no central government to offer surrender.” For the same reason, Mongols failed to conquer the medieval Rus, it…
They didn't really have castles there. Castles were pretty much impenetrable until artillery was invented. You could try to starve out the people in the castle, and that did occasionally work, but transportation networks were poor so it was hard to keep an army supplied long enough for a siege to work.
In short they would have laid waste to Europe.