Sewing friends, I have a problem and need your expertise and advice.

This is my Starfleet Science Officer bathrobe, which I have had for many years, and which is no longer in production. For obvious reasons, I want to use it until it is completely falling apart and no longer functional.

The spot at the waist where the belt gets knotted is pretty much wearing through. Otherwise, it is in fairly good shape.

I do not have a sewing machine. I do know hand sewing to some extent, certainly enough to sew a patch on and do a little further stitching to keep it in place. The closest match I was able to get to the fabric that will work is some washcloths that will kind of go with the fabric and not look too horrible, though this is certainly going to be visible mending.

I am kind of leaning toward the lighter washcloth for this particular patch, and putting aside the darker one for the day that elbow patches will be needed, which will not be too far off.

So, any thoughts on best technique? Should I fold the edges of the patch under and backstitch all around the thing, should I not fold and whipstitch around the edge? Any and all advice will be most appreciated!