are the keys that don't work grouped close together? If so you've probably dislodged the ribbon cable.....
Tosh keyboards are usually reasonably cheap and easy to find - whats the model of the machine? Usually easy to find the keyb through Googel or Ebuy.
And no you wont be able to copy to an external drive or USB stick and boot from there - the drive lettering in the registry would all be wrong
You MAY be able to get away with cloning to another drive, but if the drive is dodgy anyway you may hit problems
If Tosh didn't give you a recovery disk, you could always download an ISO of the Windows setup disk from Digital River, install from scratch and then download the drivers from Tosh, though usually theres a "make driver CD" routine on a Tosh laptop anyway (or a drivers folder you can copy)