At the adult/ alternative high school program where I work, we take students who are no longer enrolled at a "traditional" (regular) high school-- usually those who have dropped out or would otherwise drop out. They can enroll as young as 16 for high school completion, but any student under 18 must have a parent present when they enroll. Even though they can legally drop out, they apparently can't enroll themselves in a new school.
As for room and board, as RJK asked, some hotels, etc are really lax about checking those things (even around here we have one or two no-tell motels that don't want to know.) And a fake ID could help with any place that doesn't care to look too closely. For food, many minors can get jobs and some of them pay pretty well if you can get good work. Just have to watch out for laws restricting student hours, but again some places don't care and will work a minor as many hours as they can. Or, if the minor is NOT enrolled full time in school, they can work full time. These are easier obstacles to surmount, I think, than enrolling in a new school.
NOt sure if it's the school setting you want for your story, Soothing Snow, but if you're just looking to have your character finish high school, there are online schools that offer high school completion. They, obviously, can't require a parent present for enrollment, so it might be easier to get in and get finished with school. Or just have her take the GED test to be done with it. Some other options.
And for the plane ticket, I wonder if she could buy the ticket online and set up flying as an unaccompanied minor?