Looks to me like the piece that's broken is made of wood and largely intact. Use Titebond, or any other type of wood glue, to glue it back together. You'll need to clamp the piece in place while the glue sets. If you don't have clamps, you could use a 5-minute two-part epoxy glue. Epoxy sticks to everything, so be careful with it!
Once you've glued the broken piece back in, you can then use wood putty to fill in any gaps.