Okay, so no need to remove the wire entirely and replacing with a new one. That makes things much less complicated.

Patching up wires--I'll Youtube some starter videos.

I still have to figure out how I can wire it so when I turn right, the wire doesn't become so loose it dangles into the wheel. Conversely, I have to have enough slack so when I turn the left, the wire doesn't become too taut and impede a full turn.

Edit as I realize that I may not have described the new set up properly. Before the rack/basket, the hub light was mounted to the fork and the rear light was wired to the hub light. From there it was a simple path into the bike tube along with the other cabling. Now with the hub light at the front of the rack, the rear light wire has to run along the rack to reach the front hub light. I think I may have make the wire longer and wrap it around the front rack so it acts like an accordion?