In the real world, a girl who only needs one year of high school would prob. just take a GED and then get her diploma. Kids who drop out, then change their minds, often do this. If they lack in a particular skill, say reading or math, they take a quick course at the local community college or high school. Or they get some tutoring.
Enrolling in high school for one year just seems a bit odd to me. I can't recall it ever happening in the almost four decades I was a teacher.
But, if, as you say, she 'needs to' for purposes of your story, then in most states she could do this on her own at age 18. (At which point the administrators of the school would prob. question why she is doing so, so think of a good reason for this.) And at age 17 or younger, she'd need a parent to accompany her. She'd need a birth certificate and proof of residence. (In other words, she'd have to be living in the same town or city where the school is.)
Unless it's a private high school, then as long as she meets the school's qualifications and can pay the tuition, she's good to go.
A student needing to finish two or even three years sounds more plausible to me. Or if she were 15 or 16. And a high school senior can be that young, if she's skipped grades or been home-schooled.
One more thing, home-schooling throws everything I said out the window. She could have covered several years curricula in one year. I've heard of kids as young as thirteen-fourteen doing school work at 11-12th grade levels because their parents were uniquely skilled and qualified and the student was somewhat gifted. I've also taught home-schooled kids (who return to public school) and they run the range from kids who could discuss unified field theory (and its flaws) to kids who could barely read and had no idea the moon was in orbit around the Earth.