Thanks to all for your insight and suggestions!
The part that broke (aside from the 2 wires) is plastic so the Q-bond sounds like the best bet.
It's not even trickle-charging though, so ... what does that mean? It's not down to the severed mirror wires, is it? Like to complete a circuit or something? (Obviously I'm pretty ignorant in these matters.) I wouldn't expect that to be the case. Maybe I'll try taping them back together and see if that does anything for charging/starting.
(And Scooterwolf, good idea; I'll check battery connection now. Not clear on sparkplug location but can check my manual.)
Such a bummer, the bike was just out of the shop a few weeks ago and running perfectly... : (