Project Tonearm RE-Wire on Music Hall

Can someone please tell me if there is some trick to re-wiring a tonearm? I am working on my Music Hall MMF-5 Project tonearm and wondering how I can thread new wires through this thing? FYI- this table is being replaced, but not for another month and I would prefer to fix this by getting some new wires. The LEFT (White) lead was shot, and I learned the hard way to clamp it before taking off the base, so now it is no longer around to use as a feed wire. Only option I thought of was to re-wire by using two wires and solder them together to one of the current ones to feed them through. I'm I missing some type of technic here? Is it this difficult? PLease save the get a new turntable comment I"m already doing that, but I listen on a daily basis and my vinyl keeps me sane, I can't survive without it.

Pictures of my project are here. I'm not afraid to try anything on this no matter how difficult.

https://www.flickr.com/photos/124578203@N02/

Thanks