I have already done that.... I have them sign a contract. I have them pay for Sept-Dec, Jan-Mar, and then April-June.
It sounds like she is trying to make you feel sorry for her, and that some or all of what happened is your fault. Don't fall into that trap. If you have dealt with the matter in a professional manner, then you have done your part of the deal. It is time for the student to own up to their part.