I have a small hole (2" x 1") in my drywall. The problem is that it is very close to a work box, so the smallest patch I can find it 4" x 4".
Using a level, and straight edge, draw a tight rectangle around the hole. Using a razor blade, or drywall saw, cut a clean hole. A saw will allow you to remove most of the material, and the razor will cut it cleanly. Then you can use the hole as a template to cut out the patch.
Before the patch can be installed, you will need to install a backer board to prevent the patch from falling inside of the wall. You can use just about any piece of wood as long as it is thick enough to hold a drywall screw without splitting. The wood needs to be long enough to be securely attached to the existing drywall on 2 opposite sides for strength. The screws must be at least an inch away from the edge of the hole so it does not crumble. You can also put a screw in the middle of the backer board prior to installation to give you something to hold onto while it is being installed.
Then screw the patch to the backer board to hold it in place. You can then clean up the edges of the drywall pieces to fix any imperfections. Use drywall compound to fill in the cracks, and cover up the screw heads. Keep this coat very thin, and just put in enough to fill in the voids. Let this dry, and then tape the edges with drywall tape. After this, you can skim coat the whole area to blend in the patch.
My suggestion is NOT to use a mesh patch, and instead make the hole slightly larger by carefully enlarging the existing hole to a clean, rectangular shape (as explained in the above answer.)
However, instead of using wood to back-board the patch, simply cut the dry-wall at an angle (front-to-back) so that the inner hole is slightly smaller than the outer hole. [You do this by holding your cutting tool at an angle relative to the wall instead of perpendicular to the wall.]
When you cut your repair drywall, cut it the same way, making the outside portion of the plug wider than the inside portion. After you have it 'close' to the right size...test fit it again and again, until it fits in, and removes out, snuggly.
Lastly, insert the plug and use regular drywall mud to smooth over the patch without creating a lump. -- Skim once before applying drywall tape, then again after. if you have to use thick coats, then you pushed in your plug too far. -- knife out the mud, pull out the plug slightly, then re-mud/tape to seal it.
If the outlet (work box) is at a standard height (approx. 16" above the floor), I doubt the repair needs to be strong enough to hang a picture from it. No need to cut out more drywall, tape will be needed to finish the patch to keep the joint from cracking. Spackling over the smaller holes as you have done is cool, but the larger ones need taping. The hole you have only needs a small piece of tape over it. Tape comes 2" wide, cut a 3" piece of mesh tape and stick it over the hole so it overlaps all sides about 1/2", AFTER you scrape the excess spackle from the nearby nail holes you filled, sand them if you like. First coat over the repair will be a tight coat, the idea is to fill the void through the tape. OR, you can fill it flush to the wall before you set the tape, however you do it, get the hole filled and use the knife to keep the wall clean, (you can fill it by smearing spackle on the backside of the drywall with your finger to build up the opening enough to hold more spackle). Let dry. Second coat will build up over the mesh and around the outlet a bit, set painters tape over the outlet to help keep it clean. To get a smooth surface you will need to apply more mud than you think, each time you build up the amount you need, then smooth it off. Each time you apply the knife to smooth down the build up, clean off the excess you remove, off the knife. A clean knife makes it easier to get a smooth job.
You may be able to keep the repair small enough where you will not need to go down on the sides of the box, which if you are new at this, will make it simpler for you. It really should require going around the box a little to help make the repair blend in better.It will take a second coat over the first build up coat. The idea is to get just enough finish over the tape to hide the mesh, 1/16" slightly more perhaps.
Use a technique called a "california hot patch". Hands down the best low-tech technique to patch holes quickly. Works with either chemical setting "hot mud" or conventional drying type. The 5 minute setting type "hot mud" is incredibly finicky, but the 20 minute mixed with hot water is decent to work with and will allow you to have the patch done start to finish in less than an hour.