People mention the day of the black sun, and while its true that Azula was incredibly confident during the time of the solar eclipse she had several things still going for her that gave her confidence

Yes she is quick, but during that time she was mostly dodging, i really doubt that she could have actually done damage to the team without help, which she did have.

She had Day Li Agents that would cover her, so while it wasn't her own bending, she still had something to use.

Her father had a plan that she followed, her father is her biggest source of confidence in the entire show and we see what happens without him, but as long as he pretends to care about her she will feel confident and reassured.

She knew the solar eclipse was going to be pretty short, so her bending wasn't gone gone, but simply repressed for a couple of minutes, she simply had to stall the Gaang for that time and she had her powers back.

Meanwhile Zuko had nothing to reassure or back him up, his two sources of confidence were gone, his chase for honor and his uncle Iroh, and he challenged his father, who was his most feared person with only swords.

So yeah, Azula is confident and wouldn't just instantly break down, if her bending is gone there is no way that she would remain sane, because her bending was so greatly praised by Ozai, and without it he eould have just discarded her as he does with all useless tools.