This post is going to be about fixing it.
I had thought of another idea to get a light box there, but it would still require me getting a wire to my spot, which I discovered was actually quite difficult. Don't worry, there is another lighting solution in the works... more on that later.
First I decided to make more of a mess by squaring off my hole.
I then bought a 2' x 2' piece of drywall from Home Depot. I then traced out my hole by taping a piece of paper on the ceiling and tracing it out.
Then, I cut the same size piece out of my drywall piece.
I epoxied a shim piece of wood to the back of the drywall piece, like
Once I let that set, I used drywall compound to fill in the gaps. The key is to use thin coats, let it dry, and then use a damp sponge to even it out. The compound is water soluble so any uneven-ness smooths out.
After coat 4.
Ready for paint.
After the last coat dried, I used some ceiling paint I had on hand to paint over it. You can actually kind of see the compound in this photo, but I think that is mostly the flash. You can't see it in real life. While I was at it, I fixed up a couple other ceiling discolorations around the house.
So, even though I didn't really make any forward progress this weekend, at least the hole is patched up! And you never really know until you make a cut at it.