Some more questions? Are they mounted? If not take a single one and just jump it to the battery. I would ditch any relays you have in place for this set of lights. Negative to the lights with a plain old switch interrupting the positive. I would put a .5 amp or 1 amp fuse inline with the positive before the switch. Since the factory wiring seems to be 22/24 gauge I wouldn't use anything larger than 18. Keep it skinny. To much copper for such a low draw there may be a lot of resistance. We are talking LED not light bulbs. What size wire are you using. I have bought spools of wire that were kinked and had small breaks in it.
Just because it looks like it is soldered it may not be. I work on 30 year old pinball machines and I am use to re-soldering joints. Everyone has to use led free stuff now. I still use lead solder until I run out.