Student Exchange Programs

We are currently hosting an exchange student who is in the US on a J2 visa. She would like to return to the same high school for a second year, but the J2 visa does not allow it. I’ve heard she could come back on the F1 visa if she pays tuition to the public high school, but she will need an I-20 form from the public high school. Someone else told me public high schools cannot give I-20 forms!? It is all very confusing. We love our exchange student and would like to host her for a 2nd year. Also, she loves this high school and would like to avoid looking for a private school for the 2nd year. Any help is appreciated!