How do you fix the speedometer on a 1991 Volvo 740?

If both speedometer and odometer do not work, usually the pulse signal are missing. E.g. the wiring coming from the rear differential where the sensor is located, can be deteriorated (broken or shorted).

If speedometer only is inoperative, it is easiest to change the whole speedometer. To repair it, you need to be able to work on electronics, disassemble the speedometer unit, access the printed circuit board, and usually replace the leaking electrolytic capacitors on that board, as well as clean the circuit board of the electrolyte. Some other components may be broken also.