Broken door jamb - What's the easiest way to fix this?
You should drill out the hole with a large drill bit and glue in a wooden plug. Then you can drill a pilot hole into that and screw the bracket into that.

Toothpicks will work in the short term but if they are not glued then they will pull out pretty soon. Drilling out the chewed out wood and glueing in a plug will create a new place for screws to bite into. 

I would also suggest going for longer screw and making sure that they attach firmly to the wall rather than just the door trim.