You provided something for your child to wear and she chose to use something else instead - it's also an old item - if it were that precious the the owner then the babysitter should have chosen something else or even just used the one you provided in the first place! Of course you will do the best you can to find it, but you have already made the offer to replace it and if it can't be found then that's unfortunately just something she will have to chalk up to experience.