Fixing small broken area at lock plate of interior door jamb

My bathroom door has a minor issue at the lock plate of the door jamb. The main issue was that the screw area got forced in and I screwed in a nearby area that eventually broke. I dont want to change the entire jamb for it looks like a small issue. So I thought of sawing a few inches above and screw a new wood area. Want to see if there are better ideas to fix this.

Update:

Hi anybody " Dont see any solutions ?

My current solution is to saw/chisel the strike place area completely – about 1/2 inch. Put new wood block secured with screw on the sides leaving the center area for the strike place screws. Not sure if this will work. Anybody has experience if this will work ?