FYI, if you do solder it in there... please AT LEAST wear safety goggles so you don't blind yourself when solder drips down.

You can probably get it if you just use a razor to strip the remaining insulation and make sure you cover the wire back up with something good. Really, you can solder the wires if you have 1/4" of wire.

Also, getting a small amount of rosin on the wire ends will help things along and remember to heat the wires not the solder with the iron. You want to see the solder "whick" into the wires... but at the same time make sure you don't melt the connector you are right next to. Solder is drawn to hot, clean metal (the rosin cleans it).

Might be a good idea to just strip a couple of wires and practice a bit first.