Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!

As /u/jschooltiger noted in his post, this kind of question runs the risk of stranding us in the sea of counterfactuals, a place no historian wants to be. I can offer a short answer that will hopefully provide some of the information you're looking for, but one aspect I can't address and that would be relevant to your question is the factors that resulted in the fragmentation of the Mongol Empire. There are a lot of factors that enabled and limited the Mongol conquests, but a big part of 'why didn't they attack X' results from the empire eventually breaking up into smaller sections. I'm not an expert in Mongolian history, though, so you'd have to ask one of them for a more detailed account of that.

What I can say is that the Mongols did try to invade Africa, and they failed. In 1260 the Ilkhanate led by Hulagu Khan (the Ilkhanate had its base of power in what is now Iran and ruled much of the areas surrounding) demanded the surrender of the Mamluk leaders of Egypt. The Mamluks weren't too interested in this, and so the Mongols assembled an army and marched off to invade Egypt. The Mongols marched into Syria (the medieval name for pretty much all of what is now Israel, Jordan, Syria, and Lebanon) and the Mamluks came out of Egypt to meet them in battle.

The battle of Ain Jalut was fought near the Sea of Galilee and was a resounding Mamluk victory that crushed the Mongol army and left the Mamluk's as the primary power in the region. The Mamluks would use this regional primacy to drive the last of the Crusaders out of the Middle East over the course of the next twenty years, leaving them as the supreme rulers of the Holy Land (for a time anyway). The reasons for the Mongol defeat have been debated and are honestly beyond my expertise. What we do know is that Hulagu Khan left the site of the battle with a large portion of his troops before the Mamluks even arrived, leaving a much smaller force behind to face the oncoming Mamluk army. Why he did this is a topic of much debate. It was thought for a long time that he left because he had received word of the death of the Great Khan back in China and he was summoned to attend the election of a new Khan, but recent scholarship has challenged this idea and suggests instead that maybe he couldn't actually afford to keep such a large army in the field.

In any case, the Mongols never posed a serious threat to Egypt again, and without being able to invade Egypt they had little chance of reaching the rest of Africa.

I don't know of any books that deal with Ain Jalut directly, there must be some but I mostly approach it from the perspective of its impact on the later Crusades. Any history of the Crusades that covers the fall of the Crusader States will have to address it on some level. I know Thomas Asbridge covers it to some extent in his history The Crusades and it featured in Paul Cobb's The Race for Paradise.