Select a drywall patch that is the same thickness as the drywall in your wall, and is large enough to completely cover the hole. Use a drywall saw to cut the patch into a square.
Place the patch over the hole and trace its outline with a pencil. Remove the patch and use a drywall saw to make a cut from the hole to each of the four corners of the square. Cut along the penciled outline of the patch and remove the four triangles of drywall.
Measure the height of the new square hole, and the width. Cut a piece of 1 by 3 strapping with a handsaw so it is 6 inches longer than the hole height.
Apply a bead of construction adhesive to the length of the strapping. Insert the wood into the hole so 3 inches sits above the hole and 3 inches sits below the hole and the adhesive faces you.
Screw two screws into the strapping above the hole, and two below the hole. Put the drywall patch into place in the square hole. Screw two screws into the patch and through the strapping.
Cut a piece of window screen 2 inches longer than the patch and 2 inches wider than the patch. Set the screen aside.
Spread a thin layer of joint compound over the patch and the screw heads using a 6-inch taping knife. Press the window screen into the compound over the patch. Smooth it with the taping knife.
Cover the screen and the spot where the screen meets the wall with another thin layer of joint compound. Work from the center of the screen outwards toward the seams. Let the compound dry for 24 hours.
Cover the dried joint compound with a final layer of joint compound, using a 12-inch taping knife. Spread the compound 6 inches wider and higher than the first coat you applied the day before. Let the compound dry 24 hours.
Sand the compound smooth with a silicone-carbide sandpaper. Wipe the drywall with a tack cloth to remove sanding dust. Paint over it if necessary, so the patch matches the rest of the wall.