If you put off fixing a hole in a plaster ceiling, it will only get more difficult to fix. Plaster--the method of finishing walls and ceilings used before the advent of drywall--will progressively crumble and fall from the edges of a hole as moisture gets underneath it. The best fix for any plaster hole of more than a few inches across is cutting the damaged plaster back to the point where it's solid, making a patch out of drywall and installing it in place of the plaster.
With your hammer, gently tap at the ceiling to loosen and knock down all loose plaster. Continue tapping and knocking down solid plaster until you find the location of the next joists at the edges of the hole.
Mark a square cutout area around the hole, using your level and pencil. Set the level along the middle of the lengths of the joists at either side of the damaged areas.
Run your razor knife repeatedly along the marked square cutout lines, until the plaster comes off, leaving you with a square hole bordered on two sides by half-exposed joist widths. (Don't take down the lath; you should be able to see the joists through the spaces in the lath.)
Measure the square cutout area. Transfer the measurement to a sheet of 1/2 inch drywall, subtracting 1/4 inch on all sides. Cut out the piece with your razor knife.
Set the cut piece of drywall into the space. Secure it with your screw gun and drywall screws, sinking them every 6 inches or so along the joists.
Press mesh drywall tape around the perimeter of the drywall patch, so it's sitting partly on the drywall and partly on the surrounding plaster. Spread a thin layer of plaster (about 1/8 inch thick) over the tape with your drywall knife. Let the plaster dry to the touch.
Lightly sand the plaster, feathering it at the edges with your drywall sander.
Apply a second coat of plaster, making it a little wider than the first. Smooth it out and let it dry.
Sand the second coat of plaster smooth. Apply a third coat of plaster, making it thin and extending an inch or two beyond the edges of the previous coat. Let it dry. Sand again. Repaint the ceiling.
Wear a dust mask when sanding the drywall.