So I just had a fiddle with things, and put the unconnected wire to the jack with my hand and it worked fine.
So now I know I have to solder the wire to the jack. But, the original wire is now too short to connect to the jack, so can I go out and buy any old piece of wire or do I need something special?
Also, there is no sign of a previous connection between the wire and jack! (I know there must have been, but it might have fell off?) So can I just solder the wire onto the jack at any position or is there a certain area I should be aiming for?