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Inspect the drive bit that inserts into the transmission, making sure it is either still square, or still has the drive tit on the side if it is round. If the above is correct, connect your drive piece on the transmission end of the cable to a drill, and see if the speedo registers while running the drill. If no speed indicated, look behind the speedo in the dash and make sure it's drive bit on the end of the cable is not worn out. Spin the gauge drive with your fingers and see if the speedo jumps, or moves at all. If the drive bits are cool, the drill doesn't make it work, and it moves with your fingers, you need a cable (most likely).
The above tests are for normal wear. Your description of the problem indicates a possible "oops" by a mechanic.
Let me know after you do those checks, and we can speak further about the driveshaft imbalance, or slip yoke misalignment causing the vibration.
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