The flat, "winged washer" tabs must fit in the slots for the speedometer to work. A common malfunction occurs when the wings are NOT in their slots when the front axle nut is tightened, bending the wings and thus denying speedometer cable rotation. Don't know if this situation applies or not.
Solution to that problem: FLATTEN washer wings and install correctly.
Otherwise, the hub end of the speedometer cable can be chucked in a COUNTER-CLOCKWISE ("left") rotating drill; testing the rest of the speedometer mechanism. Odometer and needle should move when drill rotates.
A working bicycle speedometer should get you past inspection; don't know if a GPS receiver would pass (the DMV is most interested in the odometer, I imagine, for consumer protection upon re-sale) or not.