Drywall – How to repair this stair rail hole and screw back in

That looks suspiciously like somebody decided to cut a door through the wall and either gave up or decided it was too much work. The wall construction looks like veneer plaster, which would fit with the age of the house.

Whatever method you decide to use, you're going to have to get some more framing in there. You at very least would need to replace the stud in the middle that was cut off and either cut back far enough that you can hit a stud on either side of the hole (the option I'd go with), or put in enough blocking that you have solid edges. I'd just use sheet rock for the patch and shim it out so it is flush with the rest of the wall surface.

Hopefully there was a notch drilled in the plate, so that when a wire staple is set it will pull the wire in behind the face of the drywall. It should have been cut out large enough to allow a metal plate to cover the wire to protect it, although it is unlikely there will be any nails or screws going into the corner right there, but you never know....

Anyway, after the wire is held in place with a staple, pack the slot with drywall mud if you prefer, but it will take a long time to dry, unless you use a fast setting type and scrape that flush with the wall, do not allow that to build up or sag. Add another layer of mud for a setting bed for the tape, whether it be paper or mesh. Set the tape over the whole slot, running it into the corner and out onto the wall and ceiling. It may be a little tricky, but you can do it. After that has set add you finish coats, let dry and sand smooth.