I would be afraid to use an old sheet as backing. I have had older, expensive sheets that tore when I pulled them up to make the bed. Body oils deteriorate the fabric of sheets.
Maybe you could tell her about a forum thread that you recently read that warned against using old sheets as backing. If you can afford it, try to find Alf fabric on-line, or donate more of your stash for her backings. It would be so sad for everyone involved if the quilt fell apart.