did you try toggling the key a few times?
I know the service manual for the 650 can be found free online in a PDF file. If you can find the service manual for the 1100, the schematics for the wiring are on the last pages.
It would tell you if the key switch has different contacts for the lights and the ignition, that would allow the lights to stay on but the engine stops.
I would assume when you say the brake light says on its really the tail light? That would be a single point failure because the tail light and head light are on the same circuit. I assume you disconnected the battery to get the lights to go off. If you hook the battery back up, if the lights come on again, or they stay off, that is more information (relay hanging up or a shorted wire...)..
If you hook up the battery and the fault is still there, see if you can tell if its the brake light or the tail light - grab the front brake and see if the light gets brighter.