Not wanting to highjack the thread, but have you checked the fuse in the plug - plugs and fuses used for these obviously take an abnormally high current?
If fuse tests OK, then test for continuity from the 3 pin plug end of the lead to the plug that plugs into the car.
If you don’t have a test meter, or are uncertain about how to use it, it’s much safer to nip the cable around to an electrician to test it for you.
Kind regards,
Gareth.