Climb your ladder. Tap the damaged area with a hammer to knock down the loose plaster, until you reach solid areas. Continue knocking down more plaster until you reach the joists (ceiling support beams) at both sides of the damaged area.
Use your level and pencil to mark out a square area around the hole, with two sides of the square sitting at the centers of the two joists on either side of the damaged area.
Use your utility knife to score the plaster along the pencil lines. Cut repeatedly at the lines until the plaster comes down on the inside of the cutout square but not the outside. Leave the wood lath in place.
Measure the depth of the plaster at the edge of your cut and buy a sheet of drywall at a thickness as close as possible to that measurement. The most typical thickness you'll find is 1/2 inch.
Measure the square area that you cut out. Mark the measurement on your sheet of drywall. Subtract 1/4 inch on all sides from that measurement. Use your utility knife to cut out the piece.
Fit the piece into the space. Drive drywall screws along the two edges where the joists are, putting screws every 4 to 6 inches. Sink additional screws about every foot over the surface of the board into the lath behind it.
Lay drywall mesh tape over all four edges of the new piece of drywall so it spans between the drywall and the surrounding plaster. Use your drywall knife to spread a line of joint compound over the seams, making the line about 4 inches wide and about 1/8 inch thick.
Let the joint compound dry for four to six hours. Lightly sand the compound, then apply another layer. Repeat for three layers, let it dry, then sand it very smooth. Repaint the ceiling.