You can start these bikes without the starter button by placeing a bit of metal across the big terminals on the starter solenoid. But, the key must be on for this to work. So I would think that a stuck solenoid or a stuck button would stop turning the starter once the key is switched off.
Now, the cable from the battery to the solenoid could be contacting the starter cable somewhere.
Be careful, something's bound to heat up and burn. Also, if you've got the "newer" style key switch, up between the guages, look around there for the culprit, keyswitch-wise anyway. Those sometimes work, don't work, stay on, stay off, etc....So it's possible the bike didn't actually turn off when you switched the key off.
Also, does the red knob off-on-off switch kill the bike?
Chris in Boston