Start with the tiniest drilll bit you can. Much smaller than either side of the doll clip.
Drill a small hole into both broken pieces where they came together.
Use a small double sided screw to join them (you can just cut the head off of a screw like I did or use a real double sided screw). Screw it into the larger part first.
Then just screw the small part onto the larger part. It is best to unscrew it all once you fix it, then put a tiny bit of super glue in the holes and reassemble to make sure it holds.
Hope this helps with your broken doll stand :)