You can probe the switch while it's installed. Use a meter or test light to measure voltage from where the wires solder on, to ground. The wires of interest are white and white with black stripe, these are feeds from the battery and must be live all the time. The black with yellow stripe is the output from the switch that feeds the ignition. It should be live with the key on II (run) or III (start) positions. If you have power on the white wires but not the yellow/black, the switch is not making contact. You can also take the switch off the key cylinder and turn it directly to be absolutely sure it's not a problem with the cylinder.