I also rescue and have had this problem! Take off the entire piece of the jamb if possible. Cut off the damaged area with a miter saw. Replace it. Measure the empty area and nail in a well cut replacement piece. You do not need glue, use thin finshing nails (small head, thin). Go to Home Depot and get caulk that is both quick drying and paintable. This will fill in any gaps due to uneven cutting. It will work best if you just use your finger. When it dries sand it smooth. Also buy a sand paper piece with a foam backing (made by 3M paint area) in a fine finish. Apply 2 coats of paint blending into the damaged area. Skip the primer it saves time. Make sure Harley gets some love rescue dogs act out of fear! yelling at him will only add to his confusion! I have rescued 4 dogs and know that love,safety and consistency will bring out the best in them. Give it time in 1 year you will be amazed at the difference! Good luck!