I first thought you could drill straight through the holder to insert an LED, or you could insulate the legs and stick those through. It shouldn't be difficult, and you could even use the original wiring or just add one new wire for the whole string (in series). Series wiring for LEDs is different than parallel, tho.
If you measured the voltage going to the lamps, you could buy LEDs for that voltage.
Have you considered trying to repair the wiring you have? Usually, the problem is that the connector(s) are ccorroded and the resistance drops the voltage. High on my 2-DO list for spring, after I get my car back.
If somebody came up with a special chemical or process to reverse the corrosion, he'd be a MILLIONAIRE!! Hopefully ME.