You're not trying to launch a missile here. Go to Home Depot and grab a couple free paint sticks, a 2x2 piece of sheetrock, some drywall tape, 1" drywall screws and joint compound.
Cut the hole square, then put a couple of paint sticks in the hole and pull them so they are up against the back of the drywall. Run a screw in each side holding the sticks to the drywall. then cut a piece of the 2x2 drywall to fit the hole and screw it to the paint sticks. chamfer the edges, put on some joint compound then the tape, then joint compound when its dry and sand like hell. Nothing to it.
Ray