Lay a drop cloth under the damaged area to protect the floor from debris.
Cut out the damaged area of the ceiling using a utility knife. Cut a squared piece far enough over in every direction to expose the ceiling joists. This provides you with a solid surface to which to attach a new piece of drywall.
Measure the width and length of the area you cut out using a tape measure.
Lay a new piece of drywall on a hard, flat surface with the paper side on top. Cut a piece matching the measurements from the ceiling. Score the drywall with a utility knife on the first pass and then cut completely through it with subsequent passes over the drywall.
Fit the new piece of drywall into the corner of the ceiling and secure it to the ceiling joists using 3/4-inch drywall screws. Fasten the screws using an electric drill with a driver bit installed.
Cover all the seams by evenly applying self-adhesive drywall tape. Fold the drywall tape in half length-wise and set it into the crevice formed between the drywall on the ceiling and the top of the wall. This will provide the basis for a square corner once the joint compound is applied.
Smooth joint compound over all the seams using a 6-inch drywall knife. Apply the joint compound with the knife using smooth motions. Remove excess joint compound by holding the knife at more of angle to the surface and pulling it over the joint compound. On seams where the ceiling meets the wall, smooth the joint compound on the ceiling-side into the corner. Then smooth the joint compound on the wall-side into the corner. Use a corner of the drywall knife while applying the joint compound to create a smooth line. Allow the joint compound to dry for 24 hours.
Sand the repaired area using a drywall sanding screen. Lightly move the sanding screen over the dried joint compound to remove ridges and to blend the joint compound into the rest of the surface.
Apply a second coat of joint compound using an 8-inch drywall knife and let it dry. Sand the second application. Paint the ceiling using a paint brush and paint that matches the rest of the ceiling.