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Unless something has changed, I would think there would be an easier solution that rewiring it. Check the fuses to see if one is blown. Then, take the lighter out, clean everything up, and reinstall. If memory serves correct, the ground is where it screws together and makes contact with the frame of the ash tray door. If power supply isnt the problem, then a new one won't fix it. You said "it appears to have power, but does not" I'm a bit confused whether there is power to the existing wire or not? Your problem could be a ground issue.