Mark a square perimeter around the damaged area with your carpenter’s square and pencil. Make two opposing sides of the square land on the middle of the edge of the ceiling joist on either end of the damage. Determine the joist position by pulling down additional chunks of drywall, if necessary, and feeling inside the ceiling.
Cut out the damaged area, using a jigsaw. The two joists should be half exposed all along their edges.
Measure the dimensions of the hole with a tape measure. Mark out the dimensions on your sheet of drywall. Use a utility knife to cut the lines just to the insides of your marks (so the piece will be slightly smaller than the hole). For each cut, score across the piece with the knife, then snap it in two.
Set the drywall patch into the hole. Sink drywall screws along the two edges where there are joists, putting the screws every 6 inches with a drill and drywall bit. If joists are running through other parts of the patching area, secure those as well.
Set drywall mesh tape around the perimeter of the square. Spread joint compound over the tape with a 6-inch drywall knife, getting it smooth and flat. Let the compound set for six hours.
Spread another layer of compound with a 12-inch drywall knife, making the compound seam wider and flat. Let it set, then apply a third, wider seam. Let it dry and sand it flat with a drywall sander.