Use a utiliy knife or a keyhole saw to cut out a much larger sguare hole. Lay out the cut lines with a ruler and a speedsquare to get a nice hole with square corners.
There are many techiniques to attach a new piece of drywall. You can fit in cross-straps of 1X3, securing them with screws through the good drywall. At the hardware store, you can get these neat little metal brackets, which essentially do the same thing. Anyway, you fit in a new piece of drywall, then patch the seams using the paper tape, and drywall compound. You will need 3 layers, each trowled out thin. Use a 4" trowel for the first coat, then a 6, then an 8".
The hardest part is to match the texture. There are numerous products available in spray cans now, and with a little practice, you can do a pretty good job.
For the best appearance, I would plan on repainting the entire wall to get the best match. If the paint job is not too old, and is flat, rather then semi gloss, you might get by just feathering in the paint;