Fixing part of interior door jamb/frame
If the bottom of the jamb is loose, shim behind it and fasten with a long screw. (Use the gap of the door to set the amount of shimming. If the screw draws the jamb in too much, back the screw out and push the shims in further.)

Then, get a chunk of wood matching the jamb. I'm guessing 4-1/2" x 3/4" thick, by a few inches tall. Cut the new piece approximately to size. Don't kill yourself getting it perfect, because you'll use wood filler to cover any gap. Shim the stud behind to leave about a 1" gap. (In other words, make it so your replacement piece is set back 1/4" from the existing jamb when you put it in place.) 

Make a series of thick (1/4" or so) beads of polyurethane construction adhesive (Loctite PL Premium 10 fl. oz. Polyurethane Construction Adhesive is fairly thick and would be suitable. Most of the "liquid nails" I've used are too runny) and press your replacement piece into the glue. Get it level with the existing jamb and let the glue set. Check the reveal when the door is closed. (Stick some business cards in to maintain the gap if it's a bit tight.) A day hence, start the process of filling the gap (just about any wood filler will do) and caulking the edges where the new piece meets the casing.

If you had to have the stop going to the floor, a chunk of that could be nailed on, but I probably wouldn't bother.

Cautionary note about the PL glue... don't get it on your hands or the adjoining carpet. If you get squeeze-out when you're putting your new piece in, just let it dry and cut it later with a utility knife.