I have a puncture wound that is not healing and I keep finding bits of my shoe in the wound?

Not good. needs to be cleaned and likely antibiotic to prevent infection as shoe in a wound is very dirty. See a foot/wound specialist .
Surgery. If you were my patient, and I saw this, I would schedule a surgery quickly, to formally open the wound and clean it out, and take cultures for infection. Dr. Latva.
Seek treatment. Your best option at this point time is to find a good podiatrist and have them do an incision and drainage to remove the remaining foreign bodies so you do not get infected.
Puncture wound. See you doctor or go to a facility for care right away. Wound needs to be examined and cleaned properly.