I don't see why not-if the scooter was able to propel you by itself then it should be fine for power assist(or better) on a bike.
Were you planning to pedal this bike or just replace the pedals with the motor? If the former then you need to be careful of the Gearing lest the pedals start slapping your feet around(a freewheel pedal crank would solve that quickly enough)-if the latter, be advised that a powered, pedal-less bicycle might be illegal in many states...
If you wish to retain pedal power your first solution would work-other options include:
-independent chain to a gear on the opposite side of the rear wheel
-Rear wheel in the front fork for freewheeling FWD(beef up the drop-outs if you do this)
-friction drive directly to the front or rear tire(has its drawbacks but it easy to set up).
Also, do you know why the Scooter stopped Working? Eliminate that one component and you can likely salvage all the others for the bike(batteries, controller, wiring harness etc)...or fix the Scooter of course.
The link below has everything you'll need to get started:
Endless-sphere.com • Index page
(I lurk there once in awhile but refuse to join until I have the time to make one of my own) Also:
Browser Warning
Best of luck!