You may have bad Start and/or Key Switch relays. That's what it seems like after looking at the wiring schematic anyway. Expecially since you're bike stays running and the Lights won't come one. During normal start procedure, the Lights will cut out (relay latches for Start circuit) while the Start button is pushed. Then when you release it, the Lights should come back on. I'd swap those out for new ones and try it.
Then again, it may be as simple as a connector or you're relay box base being loose, corroded, etc. Check those first, but sure sounds like the relays may be staying "latched" for some reason.