For a split second, only a split second, Katara sincerely considers not doing anything.
Then she remembers she's not one of the bad guys. Enemy or not, a mini-version of her father or not, Azula's still a person and one who's injured. Katara can smell burned flesh and blood, and she's not moving other than twitching with the leftover of lightning. Everything until this moment has been a red flag, she fought Azula times enough to know this is not her, and the princess taking Zuko's baiting was more than enough confirmation. Azula asking for an Agni Kai seemed weird enough, shooting lightning at someone she knows can redirect it? That's not tactical genius Princess Azula.
(Katara decides to not think about how Zuko redirected Ozai's lightning at his feet, not even directly, not even close enough to injure him.
And how he aimed Azula's comet-powered lighting at her chest.)
He puts an arm in her way when she starts to jog over the princess, she doesn't even look at him as she dismisses his protests and almost sprints the few meters between them. Between what she sees as a patient in need of help and a healer that can help the patient.
As she approaches, she can see even more clearly that this Azula is nothing like the terrifying royal she got used to (the royal that, in hindsight, is what you expect from a child loyal to a parent that happens to be the worst man alive – a royal that is not so different from the Zuko that chased them, sent an assassin after them, betrayed them).
She can see her chest raising, barely, but raising. The chest piece is smoking, Katara can see the damage and can tell it was a killing shot. Katara fumbles with the buckles for a moment, her fingers burn when she pulls off the piece of armor. It's heavier than she expected, for an armor that isn't too bulky. (She heard of how agile and fast Azula still was during the eclipse, quite impressive with all this heavy armor on her).
Azula's alive only because the armor took the brunt of the impact, the actual lightning still went into her body.
Straight to her heart.
So she won't be alive for much longer if Katara doesn't try anything, and she's not sure normal water can save her. (It'll hit her, eventually, that Azula's her age.Look at what Ozai did, the feeling of life leaving someone isn't something you forget, look at what Zuko did).
The undershirt's black, and it's wet, and Katara's fingers stain with red when she touches her chest. She hoped it was sweat. She knew it couldn't be, not with how fast Azula's chi is diminishing.
Katara feels her dying.
(It's ironic how this is almost worse than with Aang.
She didn't feel him dying slowly.)
Katara knows she has almost no time at all before the body's too empty, too gone to recover. Normal water might keep her alive for long enough for a better, proper healing, but it'll have to be enough to keep her alive. Perhaps this technique won't work. Maybe it doesn't work with firebenders, with powerful firebenders to be more specific, maybe it won't work because Azula's past the threshold of how far you can be gone before you're brought back. Maybe it won't work because Katara has never done it before, and has read about it with the offensive purpose in mind.
Katara is careful, delicate, gentle, all because taking body heat can be easily fatal. The alternative is letting someone bleed out in the courtyard of their own home, that's much more cruel. Azula's breathing slows down, but it's stable, and her body stops twitching. She lays motionless, she does look dead and that might be her one chance to survive.
(Could she have dodged? She's fast, faster with the aid of her fire, she should have dodged.)
Has Azula always been this small? This sickly thin? Too fragile, like an injured bird. Too light, Katara still can lift her even if she still has parts of her armor on.
"I'm taking her to the healer's ward."
Her voice sounds almost wrong, too stoic with anger underneath. Zuko doesn't even argue, just steps ahead.
As a warrior, she should let Azula die in the courtyard.
As a healer, she has to try everything she can to save her.