In addition, even a cursory reading of the Mongol Invasion of Southern Song will show that the Song dynasty maintained few castles; at the time of the Mongol invasion...it had 3. This meant the Mongols could capture the kingdom with a handful of simultaneous sustained sieges. In contrast, at this time, there were hundreds of castles in Europe, most built to withstand years-long sieges, and most within reinforcement range of their neighbors.

Furthermore, Western Europe back then was heavily forested. Today, there are few such dense forests in Europe, but the most comparable visual is probably the thick African jungles you see in Hollywood films. Traversing this wilderness would have been possible but difficult for small groups. For an army of mounted horsemen, it would have been extremely difficult. For an army of horsemen each requiring multiple horses, it would have been effectively impossible.

Combining the lack of grazing pasture for the horses, the wild forests preventing the Mongols from utilizing their hit-and-run tactics to any meaningful effect, and the necessity to carry out thousands of sieges and blockades against a well-fed and entrenched enemy, the Mongols had no chance to invade Western Europe.