A speedometer can be repaired at home if it is making an unusual noise and if the needle is slow or does not move because of a broken gear or other component. However, any electrical problem with the unit needs to be repaired by a specialist.
If the speedometer is making a whirring sound, wipe the speedometer head to remove grime and dust and lubricate the bearing. Alternatively, remove the cable of the unit from its casing on the transmission end and clean using a solvent. Dry the cable and apply a layer of non-migrating lithium-based grease, then reconnect the cable.
If the speedometer needle is slow or does not move, detach the wires from the back of the speedometer and check if the gear mechanism or any other component is broken or damaged. Replace the gear or the damaged component with a new one. If these repairs do not work, replace the speedometer face and system.
To check if the speedometer is working after repairing, drive the vehicle and compare its speed with that of others. Alternatively, use an odometer calibration form that tests the functionality of the meter even as the vehicle stands. If none of these repairs fix the speedometer, the problem is probably an electrical one that needs the attention of a professional.