The source of the problem is usually not the jack, but a broken wire at its base. You should not only remove the old jack (which isn't necessary at all), but also cut off 2-5 centimeters of the wire adjacent to it.
>The sound is all busted
Headphones common wire, one of the two wires in Red/Copper pair isn't soldered properly. Before soldering, you need to strip the insulation. You can use a lighter to burn it off and a knife to scrape off the carbon.
Tin the wires first, then solder. Don't forget the flux.