I would advise sending it to the speedo shop. There are a number of things they will do to improve the overall quality and life of the unit. The speed cup and wheel support bushing is probably worn and they will replace it and adjust endplay. This was always a source of noise and vibration.
They also will clean, lube properly, and calibrate the rest of the speedo. They will know how to tell if the number wheel gears are worn to the point they catch, which is the most common cause of the odometer drive gear getting loose on its shaft.
For years before we started doing them professionally I used to fix the wheel slipping by sliding the shaft out while sliding in a drill bit to support the number string. I would line up the shaft to see where the pot metal drive gear would eventually be. I would take a small sharp punch and a vice. Laying the shaft on the groove between closed vice jaws I would in effect knurl the shaft by making six or seven small dimples with the punch. I would then slide the shaft back in pulling the drill bit out while still supporting the numbers. When I got to the knurled section I would squeeze it through the gear with the vice, Slowly! till the right depth for free play.
We now do it with new parts and total diassembly.
The temperature cable is a hollow tube carrying a gas to a bulb in the head. It is totally manual and will either have to be removed from the cluster or the cylinder head as it can't be separated. The speedo cable can be lengthened by removal from trans and pushing the cable through.
It seems to me that we always did this work from beneath and behind. On the worst restricted ones we had to drop the emerg brake handle and bracket.
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician