You've correctly identified an "L" bracket as that's what I used. I just realized that you don't have a tub, which makes this solution a bit tougher to use. In mine, I had the tub, and was able to drill a small hole in the tub to but a bolt through and then put a nut on the other side. The top part of the bracket was over the top of the chime, effectively holding it down. I left a bit of space between the chime and the bracket so it could vibrate.
In your case, because you just have the support bracket, using an L bracket would be a bit tougher as you'd have to drill into the bracket. I still think it could work though if you could drill into the bracket and have the chime sit where it should. The other end of the L bracket would then be just above the broken mount. This would stop the chime from being able to come off the broken mount that you have.
What I'm not sure is if you'd have enough clearance between the metal bracket and the top of the chime. I really wish I still had the game that I put this fix into as a picture would definitely help.
Hopefully, you'll be able to find the parts you need to fix it.