I haven't rewired any sewing machine lights, so thought I would let someone else answer. I thought I saw somewhere how to rewire a sewing machine light, but can't find it now. I'm thinking you are referring to the light for the one you posted about at
http://www.quiltingboard.com/vintage...e-t221491.html That looks to be a little different than some that I have seen. I personally wouldn't try to unglue the parts. I'm wondering if you can get to the fixture from the inside, especially since the wire is broken off at the base. It seems like I have seen some that the part that holds the bulb actually unscrews.
Maybe some pictures are in order.
Good luck.
Janey - Neat people never make the exciting discoveries I do.