I've taken a few apart, so I'll have a go at this...
To get to the mechanism, remove the winding knob and the screws under the leatherette, and take off the cover. Note that the screw fixing the winding knob has a left-hand thread - that is, you turn it clockwise to loosen.
Once the cover is off, you'll see the plastic counter wheel with the frame numbers - it should turn freely and spring back snappily against its stop. I think your problem, though, is the set of slotted disks underneath the counter wheel. These disks are driven by a gear train from the roller next to the film take-off spool (the top roller by the frame counter), and when a frame's worth of film has rolled past, a spring-loaded lever snaps into the next slot on the disk and prevents further winding. When you fire the camera, the body pushes a pin on the back which releases the lever from the slot and allows the next frame to wind on. Something in that mechanism train is probably not letting the lever to fall into the slot, therefore allowing the winding to continue uninterrupted right through the roll.
That's a simplified description of what's going on in there, so you could try to find the problem by removing the cover, mounting the back with a scrap roll of film onto the camera and watching the operation. If all else fails, the cheapest thing is probably to buy another used insert...
Jim