Great story. Been there done that. You'll have to do a LOT
better to top some of my stupid-isms. See the What's your biggest bonehead move sailing?
thread. Look for my posts.
I grabbed a couple of wires with alligator clips on them and connected them into the distribution panel. When I made the connection across the terminals of the bulb, up it lit.
The problem is not with the bulb, but with the socket. It's common for the terminals to become corroded. Recommendation: Haul out the ladder again, make sure that your insurance is paid up, get someone with a camera, climb the ladder with some sandpaper and clean the terminals. Make sure that photos are taken and posted to Sailnet.
Or buy a new socket.
I had the same problem with my anchor light, but 55' up. I replaced it with an LED light.