So it's that black wire that goes to the tab under the red wire...
The black wire might pull out a little so it can be stripped a bit. Flux would help soldering to the black wire strands. They tend to get oxidized over time. Maybe solder a wire onto the tab and then connect the new wire and the old wire together. Hopefully you can slid a short insulating sleeve over the joint...heat shrink if you are careful not to heat it soldering. Maybe a needle nose clamping past the heat shrink sleeve while soldering. You need 3 or 4 hands for this. 
