I'd work backwards from the jack to the rest of the electronics. Use a multimeter, set to check resistance. You're looking for 0 resistance to indicate a clean connection.
Plug a cable into the jack. Touch one probe to the shaft of the plug on the other end of the cable and the other probe to one of the wires going into the jack from guitar electronics. If the needle stays at infinite resistance, then switch to the other wire. If the needle stays at infinite resistance on both wires, then you have a bad jack. Otherwise, switch to the tip of the cable and repeat the test.
Next put a probe on each of the wires going into the jack and put all of the volume knobs to 10. The result should be the DC resistance of whatever combination of pickups are active (will vary with the switch), and should be in the 6-20K ohm range. If the jack is working properly, then the chances are you'll get zero - since you're getting no sound.
Since you're getting no sound at all, the chances are that the pickups are fine, but some component that the signal from both pickups passes through has failed. A master volume, or a master tone and the pickup selector switch would be good places to look. Follow the non-ground wire from the jack back through the circuit. Leave the probe on the non-ground wire at the jack and then touch the other probe on the other side of each component in turn. When the resistance stays at infinity, then you've found the bad component or solder joint.
I tried everything but it doesn't simply work