You won't be able to find the exact part that holds your button on. You will have to use a #15 button washer, also called a spring washer for threaded nails. Most upholstery shops would have them, but if you can't find any locally, send me a PM and I'll send you some. You may have to trim them to fit under the loop, but you can cut them with a heavy scissors.
No one lives forever, the trick is creating something that will.