Watch this video from This Old House to learn how to fix a broken door jamb.
1 Remove any nails or screws from the damaged area.
2 Nail pine 1x4 guide blocks to the doorjamb around the damaged area.
3 Adjust the router's depth of cut to equal the thickness of the poplar patch.
4 Set the router in place, with its guide bushing against the inside edge of the 1x4s.
5 Start the router and carve out all the wood from damaged area.
6 Remove the 1x4s, then use a chisel to square up the corners of the routed recess.
7 Set the poplar patch in place and mark it for cutting.
8 Cut the patch to size on the miter box.
9 Glue and nail the patch to the doorjamb.
10 Sand the patch smooth using a random-orbit sander.
11 Apply one primer coat and two paint topcoats to the repaired area.