Remove the damaged door from the hinges. Use the flathead screwdriver and hammer to pop out the hinge pins. Place the door on a flat work surface, such as a pair of saw horses. Remove the doorknob and any other hardware on the damaged side of the door.
Sand the door to prepare it for the contact cement. Contact cement needs a rough surface to form a strong bond. Wipe surface clean of all sanding dust.
Measure the door and cut a piece of luan to this size. When cutting the luan, always have the good side up. That way, if the underside splinters at all during the cutting process, you won't see it. Mark the location of the doorknob and make a hole in the luan with the drill and hole-saw bit. Place the cut luan on a flat work surface. Wipe the luan clean.
Spread or spray a coat of contact cement on both the damaged door surface and the wrong side of the cut luan. Let the cement dry until it is just barely tacky and nearly dry to the touch. Depending on the type of cement used, this can take anywhere from five to 45 minutes.
Place the luan on the damaged door so that the cemented surfaces face one another. Use the brayer and make sure that the two surfaces are completely bonded together. Replace the doorknob and rehang the door.