It's hard to tell from the picture, but this looks like the type of Jones plug with hollow pins. If so, you can re-solder each wire by heating the outside of a pin with a solder iron until the solder is molten, and then pushing the wire down into the pin. This will often eject the old (broken) bit of wire from the front end of the pin, but it doesn't matter if the old broken wire stays inside the pin. You may also want to add some fresh solder to the inside of the pin when re-soldering. Of course you want to avoid getting any solder on the outside surface of any pin.
Also FYI, the entire backbox insert (the plywood rectangle that all the parts are mounted on) should be removable without too much effort. Just speaking from my own experience, the 10 minutes or so you spend removing/replacing the insert are definitely worth it. It's much easier to do this much re-soldering at a workbench, rather than while the insert is in the machine.
- TimMe