How can I fill a drywall hole in a corner?

Orders are actually easier than flat surfaces. In hour case I would see is i could slip some “paint stir sticks’ behind the sheetrock for support and hold them in place with 2 or 3 Sheetrock screws. For cases like this I use mesh like on the right for the back and let that dry with a coat of mud on it of corse. Then 2 or 3 more coats letting each one dry between coats once you have it almost filled then paper tape in the corner and over the hole it may take a couple of strips. A thin coat of mud over the top let that dry and sand fill low spots sand high spots but with a little time you can have that hole sealed up and ready to repaint. If you have never fixed a hole there are as many different ways as there are people. I recommend folks new to tape and texture not use setting compound just standard Sheetrock mud it takes longer but almost every DIY first timer that used setting mud or Hot mud (it sets up in 15 minutes to 1.5 hours) has made a mess that I & others like realtors & landlords see when I enter the room.

Watch some uTube videos on patching holes but my advice is use normal Sheetrock mud and get at least a 6” mud knife and a hand sander with abrasive cloth or ~200 grit sand paper.

Most home stores have patch kits that would work for you. However, if you have some extra pieces of drywall around you can make your own patch.
I fnd it easiest to square off the hole with a utility knife or keyhole saw. Then cut a piece of drywall to fit inside the cutout - it doesn't have to be perfect. Then take a narrow piece of wood like a paint stirrer and cut the length slightly longer than the hole. Then hold the wood strip on the inside of the hole while you screw it in place with drywall screws from the front. Do this on each side. You can then put your drywall pieces in place screwing them to the wood strip with drywall screws.
You can then patch the corner and seams with mesh drywall tape using joint compound and smoothing it out with a drywall trowel or broad putty knife. Put the first coat on and smooth it out working away from the corner. After it dries you can sand it lightly and then skim on a second light coat to cover any imperfections feathering it out into the surrounding wall.
If needed you can buy cans of spray texturing to finish. Just follow the directions.