I don't think my answer would work well for a kid's cartoon. But I think the fact that she was broken and was not "fixed" is, in some sense, a good thing for the cartoon.
Even if you try your hardest to do the right thing, some people are going to get hurt, and there's nothing you can do about that. So I'd probably have her either stay as she is, or commit suicide.
Kind of depressing, I guess—she was one of my favorite characters. Even so, I dunno. I kind of dislike endings where everything turns out unambiguously OK, because I don't think life works that way.
I guess a less depressing proxy would be for her to lose her manic energy and become more of a sorrowful/thoughtful monk person. But that would destroy much of what I like about her character, so I dunno.
I agree that it is probably for the best that Azula's ultimate fate is left unresolved, given that people seem to have quite different opinions of her.
Personally, though, I like the idea of Azula eventually redeeming herself. It wouldn't be an easy transformation by any means - she would really have to work for it and overcome a lot of issues - but I think the series drops enough hints to suggest that she still has some core of humanity left and is not a total lost cause like her father. The end of the series has an optimistic "things are still broken, but now we can start to fix them" vibe to it, and I think an eventual redemption would flow nicely from that.
Aang energybends away her ability to firebend and she spends her days in a Fire Nation prison. Ty Lee and Mai visit frequently at first, hoping that with enough positive attention Azula might "get better", but progress is nonexistent and Azula actively tries to push them away. Eventually, the two start visiting less and then not at all as they have more and more in their lives going on, leaving Azula to spend her days pretty much alone. Years go by and Azula hits middle age like a sack of bricks. By now she has crafted puppets of Mai and Ty Lee and various assorted people and lives largely in a world of her own creation. At this stage we see her reenact some chase scene or another from the series, except with her victorious (and yes, there would be a cabbage vendor puppet). More time goes by, and we see her visited by someone who looks a lot like a late 40-something Zuko without a scar - Zuko's eldest son. She is by now on the verge of "elderly", and Zuko's son thinks that she might be some sort of font of forbidden wisdom on overthrowing his father. Her advise is that he should never, ever question or betray the firelord, that the firelord's approval is all that matters, that if only he proves he is worthy the throne will be his... He attacks her for not telling him what he wants to hear, but she is saved by Zuko's daughter. The din of combat and the siblings' argument fades away and the "camera" pans onto Azula, and then pivots so that we see things from her point of view. Zuko's son shifts slightly to more closely resemble Ozai, his sister shifts to look more like herself, and one of Azula's puppets shifts to look like Zuko from her perspective, and the melee from her view looks like they are soundly defeating Zuko. Her internal monologue pipes in: "If I only prove I'm worthy, the throne will be mine..."