In the mid 13th century, France was without a doubt the richest land of the West. Inside its modern borders there were perhaps 18 million people, and maybe more, at that time perhaps 1/5 of all the people in Europe. By contrast, all of the lands of the Rus, including modern Belarus and Ukraine, had maybe 6 million people. Let us say, instead, that the Ukraine is as populous and wealthy as France in the year 1240. It has roughly the same size and is likewise covered in arable land.
In 1240 the Mongols came out of the East and sacked Kiev, burning it to the ground, ending its days as the leading center of the Eastern Slavs. Much of Ukraine, which had since antiquity provided grain exports to Classical Athens and later Constantinople, was depopulated, and land use reverted from farming to nomadism. Lands where once indigenous farmers had grown wheat, by the 15th century had become a battleground between Cossack and Tatar. The once distinct Slavic cultural groups were subjected to Polish, Lithuanian, and eventually Russian domination.
But what if the Mongols had come upon a wealthy and populous land like France at the edge of the steppes. Would the Mongols have been able to completely over-run the territory and depopulate it as happened historically?
France had a lot of people. Surely, China had over 100 million people when the Mongols conquered it; but at the time China had perhaps 7 times as many people as France, while today China has 18 times as many people. France was probably one of the most densely populated parts of the world in the early 13th century.
France was not an empire; it was a decentralized polity with a strong local nobility. Several major dukes and counts controlled territories with populations of a million or more. Furthermore, France was Christian (as was the Ukraine) and the local nobility would not accept a heathen's right to the throne. If the Mongols seized the capital and killed and supplanted the King, the previously mentioned independent nobility would have to be compelled to submit, one at a time.
France had a lot of stone castles. This is the real meat of the question, and something that set Western Europe apart from the rest of the world in 1240. Normandy had at least 27 stone castles in 1240, as listed by Wikipedia. There are 14 listed for Brittany, 13 in Picardy, and 14 in Ile de France. Extrapolating that number out to the the whole country, area wise, gives us an estimate of at least 500 stone castles. That is not counting ruins not listed in Wikipedia or wooden castles.
Assume that the historical internal pressures affecting the Mongols continued. Batu Khan wanted to contend for the title of Great Khan in 1242, an event often credited with saving Europe from the Mongol; and there were appealing lands to be plundered in the Middle East as well. Batu never returned to Hungary and Poland, where he had been successful in 1241, because he was pre-occupied invading Anatolia. On the other hand, a much wealthier Ukraine may have been a much more tempting target...
Given what we know about Mongol tactics and successes in conquering densely populated areas, areas with decentralized local control, and stone fortifications, would the Mongols have been able to completely subjugate a France-like nation located where modern Ukraine is? Or would an intransigent local nobility and surfeit of stone fortifications have allowed local autonomy to survive for a decade or two until Mongol power waned?