I've done some "save my heirloom" projects and they are always a challenge but appreciated sooo much. Personally, I would take the quilting out and take the backing and batting off, so that you have just the pieced top. I would put a complimentary border around that to square it up, then sandwich, quilt and bind. That will also let you inspect the seams in the top and repair as needed so it won't disintegrate with the first washing. You can still use the backing and binding fabric that came with it. Good luck, whatever you decide to do.