This has worked for me:
First, make sure there are no errant bits of metal in the hole you created and follow @panzrwagn advice above. Also, cover any vent holes with tape. You don’t want bits of metal finding their way unto a circuit board.
Put a bit of WD40 (or other penetrating oil) in the hole you created. The idea is for the oil to work its way into the threads of the headless screw. Wait a while, clean out and repeat. After a long while make sure that the hole you created is as dry and free of oil as possible. I used a small piece of absorbent cloth very lightly dampened with alcohol pushed into the hole using a tooth pick or the culprit Phillips head screw driver that you used previously. Make sure that it is not so wet that alcohol will go into the threads. Now, use a FLAT HEAD screwdriver that has SHARP edges/corners and that will fit tightly inside the hole and turn applying enough downward pressure to help those sharp edges grab the sides of the hole. There should be enough little crevices on the walls of the hole for there to be enough grab. Good luck.