From Chief (its4dale) on the Kawasaki Vulcan forum:

Possible fix for a bad neutral light

The neutral light went out on my 2007 Classic.  It was still getting power so we figured it was the lamp.  Hooked up 12VDC directly to the lamp and it worked.  Those lamps have copper leaf springs that make contact with the socket.  They lost their spring and failed to make good enough contact.  Bent the copper out a bit and the lamp worked again.  File that away in case you ever need it.