i believe the actual rings are lit by leds. Leds usually has a very long life, so i'd actually check for loose wires/contacts/connection to the led first.

if nothing you do with the cable can make it come back on then you'll have to try replace the led. its not hard to mount and solder it, the problem is that you might have to open up the headlight case to get access to it. I've seen people in the states putting the entire headlight module into a heated environment like a low heat oven (can't remember how many degrees) but i don't really recommend do that cos its very easy to screw up and melt the case etc.