One of the things many people find difficult when learning about the Mongol invasions is how it kind of seems to be all over the place, and that's because it is. Something might be happening in the Middle East and at the same exact time something is happening in the Caucuses region, Eastern Europe and in China all at once. That's mostly because the Mongols fought simultaneous campaigns which sounds like a silly thing to do if you just look at it from the surface, but really it isn't. By the time the Empire reached Europe it was so large that it would have been difficult to say...bring reinforcements into China from the Pontic Steppe for example.

You have a couple of options basically. You can look at an all encompassing book about the Mongols such as David Morgan's The Mongols, 2nd edition which will not only cover the rise of the Mongols, but also the historical study of the Mongols from different sources. This is good because if you're interested in the Chinese perspective, you can just read that part and skip over the European sources if you'd like. This book covers administrations and institutions as well, and also breaks down the Mongols in different regions, and this is where it becomes useful to you, someone that's looking to learn more about a specific region.

Section 5 of this book is titled The Mongols in China and it covers not only the Song bit that you're looking for, but the Mongols in China in general. It will talk about the reign of Kublai Khan, theology and Mongol rule in general.

The Mongols, 2nd edition by David Morgan

Keep in mind that virtually all Mongol books are going to mention China in general; however, if you want to get down to the details I got another book for you. Defending Heaven: China's Mongol Wars, 1209-1370 by James Waterson. I like this book particularly because it's the 'mysterious' part of this whole story. To the West, China was basically another galaxy, so everything that was going on over there was not well known for a very long time. This book covers not only the constant struggle between the Mongols and China, but the significance of this struggle. In 1251, Mongke became the Great Khan and assigned the task of invading the Southern Song to his brother Kublai, this book covers a lot of that.

Defending Heaven: China's Mongol Wars, 1209-1370 by James Waterson

I hope this helps.