Broken stem, how to fix?
Just replace it. The part is called quill stem, it is not threadless-compatible, and it is still widely used. The only type of quill stems I heard of have 1 inch diameter, so you can just find one that is of the same/similar height and length.

The simplest is to take it to a local bike shop and they will do it for you. If you want to replace it yourself, get a wrench that can undo those two bolts at the top of your picture (hard to tell, but I think it is something in 12 - 15 mm range).

Unscrew the stem from the frame and from the handlebars; grease the new stem's shaft that goes into the fork with any sort of machine oil/grease you have; then screw it back into the fork and attach the handlebars.

It may happen that the fork-stem surfaces have rusted together if no grease was used at installation time. In this case, you will have a hard time removing the stem. Look for advice on this site how to help it.