OK.
Get silver solder if you can as it makes better electrical contact and the tin-lead is not good fume-wise(lead) to have in the air.
New jack will have no wires.
There are two types of headless Corts/Steinclones. One has a battery and some sort of preamp(ACTIVE) and the other does not(PASSIVE). Also, the one with the preamp has three screws on the back cover plate.
The jack is one of the barrel type on the ACTIVE version(has a battery compartment and an LED with switch on the front) and uses either needle nose or a large flathead screwdriver to get off. The barrel jack looks to be a bit more work to get out.
The jack on the PASSIVE one(no LED or battery, two screws on the back cover) is a "normal" jack.
I believe this is the one you have. If not let me know, the other one is here as well.
Unscrew the nut on it and then, using care, the jack pushes into the control cavity and can be removed from the back with wires still attached.
The wire soldered to the back of one of the controls and to the tab touching the inner ring is the "ground". The one going from a tab on the control to the tab connected to the tip of the jack if the "postive" or hot. With new jack in hand, unsolder the old one, that is unsolder the wires from the jack. Now resolder them to the same tabs on the new jack.
Wife's got the camera in her purse out shopping!(She had it for Christmas pics at the school today). Once she gets home, we'll get pics of the bass jack job.