That is a twist grip shifter, the number of speeds will be on it. Just replace it with a Sunrace or Shimano of a similar mumber of speeds. It is not worth trying to dismantle and repair them.
Thanks for the quick response. I've been fixing bikes since I learned to ride, so my instinct is to try and repair it. Sounds like a possible lost cause!
I agree with George that the shifter is likely not repairable. On that price level of bike, most of the internals are pretty cheaply made and are not intended to be repaired.
