Much of the problem IIRC was that they were a predominately Turkic family from the north. That was advantageous for conquering China, but once it was a single state the "real Chinese" wanted to be running things.
This isn't like the Mongols or even the Manchu, both of whom had large populations and geographic regions distinct from the Chinese. That gave them a fall-back position to work from and a distinct group that needed them to remain in power to protect its own interests. The Turkics of Northern China had thoroughly assimilated into the upper classes and were thus "safe." Ironic, given that they weren't seen as being Chinese enough to rule a united China, but there you have it.
That's part of why it didn't last in OTL, mind. I don't mean to suggest that it was impossible for them to hold on, or even unlikely for them to do so.