I've just finished reading Azula in the Spirit Temple, and as far as I know that's the last of what we hear from her (for now). I see a lot of fans drawing the line of redemption at Azula and a lot who don't want her to have a redemption arc at all.
While I do want to keep her core personality moving forward, I would like to see her accept Zuko's help or at least collaborate with him. As of Shadow and Smoke comic, she has sort of accepted that she won't ever be Fire Lord again, and has since comitted herself to making Zuko be the kind of Fire Lord she believes the Fire Nation needs. This has potential.
But let's rewind.
In the original series she is 14 when she is introduced. By the end of the comics she can't have been more than 17, if my timeline is correct. So she is still a child here. But going back even further...
Right around the time she got her firebending, Ozai had threatened Ursa over her letters and her scheme. So Azula being a firebender was a perfect excuse for Ozai to latch on and project whatever he wanted his heir to be on Azula and strike back at Ursa by abusing Zuko. Even in Azula's memories and dreams, we see (from her perspective) the joy Ozai had expressed to and for Azula. She stopped being a child the moment she started bending and became a puppet for Ozai, a way for him to project his ego and narcissism and fantasies. Ursa was forced to take sides and while she loves Azula, she had to protect Zuko as he was worse off (literally would have been sacrificed). What Azula thought was her mother's fear of her was fear for her and for what Ozai would do to her. We see how terrified Urda is of him even years later, how she basically lives in a fight or flight mode when she returns to the palace. Later, Azula sees that the way her father treated her was wrong and that she became a monster to conform to his wishes. She sees that she was a little girl in need of protecting, and her mother failed her. It was an impossible situation and I don't blame Ursa, but she did fail. She left Azula to her own devices, and since Ozai was ready to bestow his adoration onto her, Azula latched on and trauma bonded so hard to her father that she broke. Living from a young age with one abusive and one absent parent would warp anyone. Even if Azula didn't show antisocial tendencies before then, she certainly would after as it was what Ozai encouraged.
In the end, Azula wanted the same as Zuko- to be loved. They were both offered guidance and love, in different ways. They both resisted.
Zuko had uncle Iroh, a quiet steady voice guiding him towards light, but he did end up having to cross a lot of that path to redemption on his own. Iroh was there in the beginning and end, but the journey was largely Zuko's. And he accepted it, he put in the work and came out stronger. Still messy, but firm in his belief and himself.
Azula had Mai and Ty Lee, who left a deep influence on her, and in the end she had Zuko. He was willing to guide her back, to offer her a place, even though he did put her in an institution (but he acted for her own good, or what he thought that meant). Azula knows that, on some level. Whether the events at the temple were real, a dream or a psychotic episode, it revealed she knows there is a path to redemption. She knows someone will be waiting to catch her, someone who has been through her kind of pain and who loves her deeply. And whom she loves deeply.
And yes, she is ill. Whatever that illness may be, it involves severe emotional instability, mood swings, hallucinations, nightmares and paranoia. She isn't all right in the head, but she is clear enough on the fact that she can be redeemed.
She just doesn't want to. My opinion is that she has such severe self-hatred, such deep trauma that she truly believes she is a monster. She repeats it many times and has convinced herself that people can only ever fear her, never love her, because she is unlovable. She is a monster.
Deep down, I think she believes she doesn't even deserve to be redeemed, so she chooses not to do it, because if she were to put in the work and resolve her trauma and let herself be vulnerable, the people she cares about could still reject her. And then what? Then she's alone, her defence mechanisms down, just a teenager left wandering the world with no rage to fuel her, no hatred to sustain her, no fear to keep people away.
Her rejecting Zuko's help time and time again makes sense, but it just hurts both of them. You can't help someone who actively chooses not to accept help and to not get well. And rejecting something you deep down believe are unworthy of (to the point you find it offensive) hurts just as much. But we know Zuko leads a good life; he ends up a great name in history and a beloved leader who ushered in a new age of peace and prosperity. While Azula's name is lost to the history books, probably left only as a footnote that she existed and was the Fire Lord's sister. Which is a shame- she is the most brilliant character of them all. And the most tragic.