Pull down enough of the damaged plaster to access the ceiling joists above it. Mark out a square area around the damage, using your level and pencil to mark it. Make two opposite sides of the square line up with the centers of the two joists on either side of the damaged area.
Cut out the square, using your jigsaw. The cutout area should be bordered on two sides by half-exposed joists.
Measure the square with a tape measure. Use your utility knife to score and snap a piece of drywall to those dimensions, minus 1/4 inch on all sides.
Lift the drywall patch up to the hole. Secure it to the joists on either side of the hole, using a screw gun to sink drywall screws every 6 inches along the joists. If additional exposed joists are running across the middle of the hole, secure the patch to those joists as well.
Lay mesh tape over the edges of the patch. Spread patching plaster over the tape with a wide drywall knife, getting it smooth and flat. Let the plaster dry to the touch.
Lightly sand the plaster with the sanding pad, then add another layer, making it slightly wider and smooth. Let it dry, sand it, and add a third layer. Let it dry overnight.
Sand the final layer of drywall so it's completely smooth.