Mark a square around the damaged area, using your pencil and carpenter’s square. Make two parallel sides of the square run along the edges of ceiling joists, pulling down enough of the damaged plaster to assess the joist locations.
Cut out the damage area with a jig saw, running the saw along the square that you marked out. Cut out both the plaster and the lath behind it, so there’s a square hole in the ceiling. Two parallel sides of the hole should run along the centers of joists.
Measure the square hole, mark out the measurement on a sheet of drywall, and cut out the square with your utility knife, scoring and snapping slightly inside the lines to make the patch a little smaller than the hole.
Set the patch in place. Sink screws along the two edges that are sitting on the joist, putting one screw every six inches. Also sink screws into the any other joists that that are behind the patch, between the two outer edges.
Press mesh drywall tape over the border of the patch. Spread joint compound over the seams with a drywall knife, making it smooth and flat. The compound should be a little wider than the tape. Let it dry about six hours.
Spread a second, slightly wider coat of compound over the seams, again keeping it flat and smooth. Let it dry, then apply a third layer. Let the third layer dry, then sand it smooth with a drywall sanding pad. The ceiling is ready to be repainted.