Without walking you through the wiring diagram and checking for power outputs and grounds checks, the problem very likely lies with the ignition switch.
When the key turns a series of electrical contacts are formed passing electricity to various vehicle components. When shutting the car off they are broken.
Either the ignition switch is failed or a B+ wire has shorted/melted to the wire powering your instrument lights. Tracking down a short can take hours, even for a professional.
If you would like to attempt the fix, replace the ignition switch first. If you think you can trace the short you will need a test light with long 10' leads and we can assist you with it.
Saturday, July 29th, 2017 AT 7:31 PM