How do I check to see if a mechanical tachometer or speedometer is functioning?

Using a known speed rotating device (e.g. an electric drill that turns in the proper direction to cause the speedometer or tachometer to indicate), attach it to the drive end of the flexible shaft.

Turn on the rotating device.

The speedometer or tachometer should register.

If it does not register, it is likely that the core of the flexible shaft is broken.

Replace the core (or entire shaft assembly) and repeat the exercise.

If the speedometer or tachometer registers, function has been ascertained.