The Earth King's Court
So we're talking like advisers and varying warmasters and important military generals. You can use the ones that were presented in canon, or just make up your own, I don't terribly care. Knowing Asante's tendencies and knowing that he's familiar with the mess Long Feng made of Ba Sing Se, I wouldn't be terribly surprised if he decided to completely remove the former Earth King's court and install his own.
The Dai Li
Now, of course, the Dai Li are still somewhat important to the city, and Asante is still a member of it. Formerly, given he was the Earth King and higher in rank than they were, Asante de-facto lead them, but more recently, Asante's given command of them to Toph Beifong. They will eventually become the first group of metalbenders, and will be acting as elite guard to Fire Lord Zuko later on.
This want doesn't really have anything solid to it, it's really more like a loose idea. But I kind of imagine Asante somewhat treating his swordmaster as something of a parental figure, and I thought it might be really cute if he somewhat leaned on his swordmaster as he figured out this whole... king stuff. Because we don't make farmers kings, but here he is and he doesn't understand how court stuff works and the last thing he wants to do is start another war. So maybe his swordmaster has a better idea how this works than he does and kind of acts like his guiding light so to say. This role's got a lot of potential I think and could be really interesting to play.
These are all just rough ideas I've got, you're welcome to flesh them out and stuff. c: I'm not really married to a full-on concept so I tried to leave as much wiggle room as I could.