I'll be fitting the cathode to a permanent +ve, with the anode to the +ve that loses power to the radio when the key is removed. That means the -ve will be provided via the residual earth through the radio when the key is removed. Hopefully.
I wouldn't rely on that working, the chances are pretty slim. What I would do is connect a small relay (only needs to be able to switch about 50mA) to the switched 12V that goes on an off with the ignition, while the NC connection goes to permanent 12V. The common would then go to a resistor if the particular LED you have needs it (there are LED's that have flashing circuits and run direct off 12V) then to the LED and ground as per the attached schematic.
When the ignition is on, the relay pulls in, disconnecting the power to the LED. When off, the LED is powered.