doorsframewood
I finally hung these doors I found at a second hand store. I am a total amateur so please hold back your giggles. No need to explain how this happened, it was a dumb DIY idea. I just need a fix in terms less technical. Is there a way to cut this out and just replace the damaged wood?
Rot needs moisture to get started, so it's unlikely that an interior door frame would be rotten without you seeing other signs of water damage on the walls, floors, ceiling etc. nearby.
A quick test for rot in wood is to try to push the point of a pick or the blade of a small flat-head screwdriver or into the wood: if it resists, the wood is sound and if it pushes in easily, it's rotten.
If you're really concerned that the frame is rotten, you should think about replacing it: it would be very hard to be sure that you've covered the entire rotten piece with hardener just by going through the screw holes.
Otherwise, I'd just go with my advice from the question you linked to.
There should be a Striker plate much like the ones on a standard doorway. 
Best Answer
I think the easiest thing to do would be to remove the doors and sand it down so that it is all flush and even. A belt sander would make quick work of this it can be done by hand too if you're willing to put in some sweat. After sanding it, chisel out material for the hinges, prime, paint and re-hang the door.
It's hard to tell for sure but it looks like the hinges are recessed too far into the door which might be why it didn't close properly. The hinge should be flush with the edge of the door. Place a couple washers behind it to bring the hinge out flush with the door.