If you are not doing the quilting, let it go. Yes it will wear out quicker but a new back can be put in later if it does. You might mention that you don't know how much life the old sheet has left in it and bring another sheet for a substitute for her to use or not use at her discretion. She could use the Alf sheet for pillowcases. You should be able to find a clearance sheet at a reasonable price at Walmart and just gift her it.