Measure the height of the hole and cut a piece of 1-inch by 3-inch wood strapping that is 6 inches longer than the height of the hole.
Apply a thick bead of construction adhesive to the front of the wood strapping.
Carefully slide the strapping into the hole with the adhesive side facing you. Pull the strapping towards you, and drill two screws into the top and bottom edges of the board through the drywall. Stagger the screws, and drive them deep enough so that they're just below the surface of the drywall.
Measure the size of the hole and cut a patch to fit out of a scrap piece of drywall with a utility knife.
Press the drywall patch firmly against the strapping and drill two staggered screws through the drywall patch and into the strapping, driving the heads just beneath the surface of the drywall.
Apply a thin layer of joint compound over the drywall patch, the screwheads and the seams along the patch with a 6-inch taping knife.
Press a piece of window screen a few inches larger than the patch into the joint compound with the taping knife.
Place more joint compound over the screen. Work out from the center of the screen to the edges so you don't wrinkle the screen, and wipe away any excess joint compound with the trowel. Allow the joint compound to dry overnight.
Apply another thin layer of joint compound over the patch, this time with a 12-inch taping knife. Allow the joint compound to dry.
Sand the patched area with a fine silicon-carbide sandpaper. Sand just enough to smooth the surface of the patch; you don't want to sand deep enough to expose the screen.
Wipe the dust off the wall with a dry rag.
Add a coat of primer to the patch, then paint over it to match the rest of the wall.