The Mongol empire covered nearly half of Eurasia. I understand its fall had something to do with political infighting and a fracturing of its territory, but what prevented each of these new countries to go out and conquer its own empire? And after the empire fell, what prevented it from recovering and once again conquering Europe and Asia?
You are correct in your assertion that infighting occurred after the death of Genghis (Chinggis) Khan. This map is mostly correct. I've used it before to demonstrate this point. You'll note the different regions: Khanate of the Golden Horde (Kipchak Khanate), Chagatai Khanate, Ilkahanate, and the Khanate of the Great Khan. Also helpful will be the lineage of Genghis Khan. This chart is a little fuzzy, but I think you'll get the idea here. The problems that lead to the fracturing of the empire are much more complicated than this, but for the sake of brevity.... I wrote extensively on the divisions of the Mongol Empire after Genghis' death here.
In short, Jochi was given territory in the west, which turned in to the Kipchak Khanate with several hordes of varying metallic characteristics. The last Khan from the Golden Horde was dethroned in 1783. Chaghatai's khanate lasted through the 17th century. Ogodei became Khan, then Guyuk, then Mongke, then finally Kublai. Hulegu was sent to deal with the Mamluks in Egypt and established the Ilkhanate which lasted until 1335. The youngest son, Tolui, should have inherited the homestead, but died before he could inherit the homelands. Kublai took the native lands of Genghis Khan and proceeded to continue to conquer China until the Song dynasty gave up. The Yuan dynasty lasted until 1368.
We can conclude that the Mongol Empire did expand in to Europe and Asia after Genghis Khan's death. The Golden Horde faced the Kievan Rus), Hulegu created the Ilkhanate in the Levant, Chaghatai ruled over the modern day 'middle east', and Khublai controlled most of China, parts of Siberia, and Mongolia. Kublai also attempted to invade Japan and sent armies in to South East Asia.
It is probably permissible to conclude that the various Mongol Khanates did not coalesce in to one large empire again because everyone wanted to rule. Only twice in the history of the Mongols were the various tribes united. Genghis Khan was the second person to succeed in uniting everyone. I doubt anyone could have done it again after him.
A couple of books you might look in to are below:
Morris Rossabi's Khubilai Khan: His Life and Times, 1988.
Jack Weatherford, Genghis Khan and the Making of the Modern World, 2004.
René Grousset's The Empire of the Steppes: A History of Central Asia, 1939.
Timothy May's The Mongol Conquests in World History, 2012.
David Morgan's The Mongols, 1986.
And I haven't read Jack Weatherford's new book on Genghis Khan, though I see he is doing an AMA on November 1st. You might ask him some questions about this subject as well! I'm sure he has loads of information!
EDIT: Formatting