Main question is if there's any electronics in the plug itself. (Would probably be voltage regulating circuitry)

If not, just cut it off and connect the wire that went to the center pin to +12V. Constant, accessory, with or without switch override... However you want to power it. Run the wire that went to the outer prong(s) to ground. That's it...

If there's electronics in there and you're not sure what they do - solder wires to the center pin and side prong, connect other end of wires to +12V and ground as above. Cover plug with shrink tubing/electricians tape and shove out of sight into the dashboard somewhere... This second alternativ is useful also if you don't want to open up the plug.

/Michael