Attach the Eagle back onto the ring. Bend the ring back together to close it off. It wasn't soldered at all, just bent into place to begin with.
That's it. You're done. You can stitch it by hand if you want. In fact, the side stitches are better done by hand to assure that it's a good stitch. Well, maybe you're better with a machine. The red thread does look a little weird on the purple background, but it looks just as odd on the BLUE background of the original medal. Again, hand stitching would have made this look a little nicer. I am not patient enough for hand stitching in most cases.