A big part of the problem is that the connection is just a physical connection where two different types of metals are in pressed into contact leading to galvanic corrosion. Here, the wire was corroded far back *inside* the insulator so I had to strip it back quite a bit and install a pigtail to extend the cable back to it's original length. Interestingly enough, I found an [instructable](https://www.instructables.com/id/How-to-perfectly-clean-wires-in-minutes) on how to clean such corroded wires but I didn't think to do something like that at the time.
With that little bit of shiny copper wire I was able to properly solder it to the bits of metal that makes contact the lightbulb. I also cleaned up some corrosion with sandpaper and sealed it up better with some caulking so not as much moisture would get inside (hopefully.) I suspect I won't be having this issue again for a while. (Fingers crossed.)