Too short is a much bigger problem. I would think you can shorten it and drill a hole at the appropriate place to mount it.
The missing wire is more problematic. I might try a small nylon washer that would fit in the base of the socket. Take a length of wire and create a "bulb" of solder on one end. Thread it through the washer and then the back of of the metal socket. The bulb will hold it in place. That's basically how a lot of prewar sockets were made but they used a fiber washer as nylon was invented/perfected until the late 1930's.