The parts contained in a computer speaker system are the crossover (separate frequencies into applicable cones), the amplifier, the drivers and the cones. If the system won't power up, it's probably the amplifier and possibly something with the crossovers. This will have nothing to do with the speakers themselves which only are composed of the drivers and cones.
With regards to the knob, if you broke the knob and only the knob, then you'll be fine, but if you broke part of it inside as well, then you might be in for some trouble.
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