You are correct about the green wire needing to be soldered. Headphone wire is a major P.I.T.A. (lol). If you can get some fine sandpaper rub the end of the green wire to get some of the coating off. Tin the end of the wire and hold it with some tweezers (it will get hot quickly) and solder it to the headphone plug. Sometimes holding the wire against doesn't make a good enough connection to test, due to the protective coating on the wire. The string I believe is there to provide some protection and flexibility to the cable. Good luck!