Speedometer unstable

My speedometer is unstable, jumping back and forth approx 10km/h / 6mph... and when I accelerate hard it fan "fly" higher and then come down.
Is there any why to improve these, or is the part inside "wear and tear".

Best to try a fresh cable before pulling the speedometer apart.
Don't use a cheapie generic cable as they tend to whip.
Liberally grease the cable before installation, I use moly CV grease, but there's prolly something more suitable.
If that doesn't cure it, it may be that the magnet spindle is loose.
This can be a tricky fix as it involves prying the clock band off gently, which takes forever.

This is inside a tachometer, but it's the same mechanism, the end of the axle holding the rotating magnet is flared. If there's too much play in-and-out, or colinear to the axis of rotation, the needle will jump. You can tighten it with a pin while supporting the axle at the other end. Dunno how to lube these shafts. Perhaps let it soak in hot grease.

Dont think there is a magnet in there. Those old speedo's have an oil cap in them which leaks out over the years. The fuller and higher the viscosity of the oil the more direct the drive to the needle. Its not a part which can be easily fixed and requires specialist tools to fix.
Having said all that, 99% of bouncy needles is cable related. Make sure the cable has no tight bends in it and is well lubricated.

Thank you

The cable was new before the season started, maybe I shall try a new and lubricate it...,

I`m not convince a new cable will solve my issue, as I said above I installed a new cable before the season stated the problem was there from first mile.
The bike was new for me then, so I have never seen it working good..

The cable was maybe not greased from factory ?
The cable was not OEM properly...

So these two variable can maybe be the issue?

Find a new OEM wire and grease it a lot with Teflon spray?

Have you performed the normal lubrication maintenance on the cables and the rest of the bike? If not, I would simply take a look at the Kawasaki Service Manual and follow its procedures. Ed

The speedo has no direct drive from the cable. Its either driven by oil viscoity or magnetic field depending which type you have. The needle is damped by a clock type spiral spring. So unless any part internally is damaged, which is unlikely, it will always read progresively higher as the the drive "field" or oil friction increases as the cable spins progressively faster.
For the needle to bounce about the cable must have a tight point which causes eratic turning. As it comes to the tight point it slows and twists the cable, as it passes it it it untwists the cable quickly back to its resting state. Much like a coil spring.
Things that can cause a tight point in rotation are,
Frayed or sticky cable. Frayed cable means a new one. Sticky cable needs cleaning and greasing.
A tight bend in the routeing. Reroute it as straight as poss.
Poor quality cable where the inner is too small diameter for the outer which causes cable slapping. New cable required.
There not expensive so suggest a new one grease it and pay carefull attention to routing.

Thank you for all the information!

The is parked for the season, I will purchase quality cable and grease it.

I’m planing to use teflon spray as lubritate, is that a good type of lub to use?

All the cables was new before season started, My plan is to lube everything during the winter.

Dont know if teflon is any good for that application, never tried it. I normally just take them off, take out the inner and clean it. Squirt oil down the outer until it runs out the bottom. Grease the inner cable with general grease and then re assemble it.
Do it at the start of every season or twice a year if the bike is used year round.

Mechanical speedometers are driven by a spinning magnet passing torque to an aluminium cup.
The rotating magnetic field sets up eddy currents which create a magnetic field around the aluminium cup.
Works the same way as an old electricity meter.
There is a small oil damping pot on the needle and a return spring. Pencil is pointing at it on a Nippon Denso tach.
I have no idea how to refill the dashpot on a kz Nippon Denso spindle if it has leaked out.
The speedo is calibrated using a machine to demagnetise the rotor rather than increasing the return spring tension.

Thank you to all for the photos.

Is there anywhere/anyone who rebuilds these meters? Between age and arthritis my fingers aren't as nimble as they once were ( ).