Erm, a power socket with no power isn't a power socket - its a socket
I had exactly the same problem though. The problem was with the socket rather than the supply to it. Easy I thought - just remove the socket from my parts car and install in the road car - Nah! To remove the socket, you have to remove the panel it is fixed to. Because of the way all the panels overlap, you actually have to remove the panels above it and that involves removing the radio and the rectangular metal housing the radio fits into - that was a right bugger. After lots of swearing and cutting up my fingers on sharp pointy bits on it, I literally just scrunched it up to remove it.
Other than that, removing the panels wasn't difficult at all. I then ended up with the panel with the power socket (cigar lighter) attached to it. I then set about removing socket from panel - Nah! There was no way of removing the socket but keeping the retaining plastic bit intact. I got it out eventually, but if I went through the same process on the road car I would have had a radio rattling around because I'd smashed the housing and a socket that was loose and falling about because there was no retaining clip.
So I decided not to go through that process. What I did do though was to release the gear stick gaitor and pull it up. This gave me an inch or 2 or space to get my fingers behind the console panel to remove the power feed from the socket. I then fitted the feed to the socket I'd removed from the parts car, connected up the power cable to the device I was using (CB Radio), wrapped it all in bubble wrap and cellotape to stop it shorting, left it resting on the "transmission tunnel and put the gear stick gaitor back.
I have subsequently had the gaitor back up to swap power cables to my dash cam and use the rear power socket for my CB as that's used less.
I presume there must be a way to remove the radio housings because people have removed whole dashs to swap out heater matrixes, plus I presume you could buy a new socket that would come with the clips. That being the case, it would be a relatively easy job and the panels all do just screw back together.