Speedo cables are lubed by the tranny fluid, If you look at the actual internal cable, you will note a spiral winding that when rotated pulls fluid up the cable to the dash. A gasket at the dash seals it from leaking. Usual cause of the problem is lack of tranny fluid changes, things gum up and away you go with pulsations. Cold makes things thicker. Eventually breakage. Reason I know this is back in the 60's there were a lot of GM prods that peed on you shoes because the cable seal at the dash was a POCrap. What a mess. Nothing like a carpet soaked with stuff and a set of ruined shoes and sox as well. Anyway, a fix that MAY work is to get a wide mouth container, undo the cable from the back of the dash, stick it in the container and drive away. The tranny fluid will push up the cable and hopefully blow out the thick crud that usually causes the prob.

The other cause is often a crimp in the cable - rocks, engine swaps, stupid human tricks etc. Another that used to work was the outer cable was steel wrapped and it just rusted up.

Anyway, look for damage to the cable first. Then try the container trick. If ng on that, get a replacement cable.