Restoring exterior door frame at latch?

Tenants have broken out wood where an exterior door should receive the doorknob latch. There's nowhere to attach the strike plate.

What's the right way to fix this? The depth of this piece of wood appears to be 1.5 inches, so it's probably a 2×4 or 2×6.

Do I cut out a few inches of wood and replace with 2×4? Do I try to rebuild the section with wood putty, or Durham's Rock Hard, or something like that? I fear the screws for the strike plate wouldn't hold if I tried that.

This seems like a common enough repair, but I don't want to have to come back to this over and over due to improper technique.