Go up to your attic. Remove the insulation over the damaged area. Place a drywall saw against a ceiling joist on one side of the damage. Poke the drywall saw's tip through the drywall. Repeat with the ceiling joist on the damaged area's other side. Use these two holes as reference marks when removing the damaged area.
Go down to the room below the damaged area. Hold a level against the ceiling. Place one of the level's edges about 1/2-inch past one reference hole. Use the level as a straight edge to draw a line along the wall. Place the level 1/2-inch past the other reference hole and draw another line. Use the level to draw two more lines connecting the lines you just drew. When complete, the damaged area will have a square marked around it.
Hold the level against one of the lines. Use the level as a straight edge while you cut along the line with a utility knife. Repeat with the remaining three lines until the square is removed.
Measure the square's dimensions with a tape measure and transfer these measurements to a new drywall sheet. Cut the square out with the drywall saw. Place the square into the hole. Check that it fits snugly.
Secure the square's edges to the exposed ceiling joists with drywall screws using a drill fitted with a Phillips-head drill bit. Sink the screws slightly below the drywall's surface. Lay a piece of adhesive mesh drywall tape along the seam where the square's four edges touch the surrounding drywall.
Scoop drywall mud out of the bucket with a five-inch taping knife. Hold the knife at a 25 degree angle with the edge resting on the mesh tape. Spread a thin layer of drywall mud over the tape by dragging the knife. Scoop up more drywall mud as needed.
Sand the first layer of drywall mud with a sanding sponge once the mud is completely dry. Do not sand down to the mesh tape. Stop sanding once the first layer is smooth.
Apply a second coat of drywall mud the same way you applied the first layer except use a 10-inch taping knife. Wait for this layer to dry before sanding it smooth with the drywall sanding sponge. Apply a third coat with the 10-inch taping knife. Feather out this layer's edges so that it blends into the surrounding drywall. Sand the third coat once it is dry. Clean the dust off the patched area with a damp cloth.
Paint a primer over the patch and drywall mud. Allow the primer to dry. Practice spraying texture onto a piece of cardboard. Hold the can at different distances from the cardboard. Spray over the cardboard quickly and slowly. Hold the cardboard up to the ceiling to compare textures until you find a match.
Spray the primed ceiling with the spray texture using the technique that most closely matched the texture of the surrounding ceiling. Wait for up to four hours for the texture to dry before painting. Go up to the attic and lay the insulation back over the patched area.