She can't even remember making her own koseki? Strange and very unusual, I think. One easy way to find out is to have her ask her parents to check their[/i] koseki and see if she is still listed on it.
Another way, less easy, is to contact her home office where she would have had it made, and inquire, perhaps even ask for a copy.
As far as getting a driver's license is concerned, does she need a koseki or juminhyo for that?
If you two are planning to live with or near her parents, contact (or have her parents contact) their ward office and inquire about what is needed for her to establish residency (permanent or otherwise) there.