Hooking up the Speedometer
 

After driving around for some time now in my Cougar, the inactive speedometer is getting on my nerves. I imagine that when they installed the new engine/tranny they just failed to hook it back up wherever it hooks up.
My question is, where does it hook up? Is it on the transmission somewhere, involving lots of work taking things out, or is it a more simple fix, hopefully in the engine compartment?

What year is your Cougar? Some use two Vehicle Speed Sensors (VSS), both near the output of the transmission, one feeding the Powertrain Control Module (PCM), the other the speedo/odometer. Other models use a sensor built into the speedometer cluster.

If it's running and shifting normally you're probably just dealing with one loose or disconnected harness. Look around the transmission output. If you see two connectors one of them is probably loose. Unclip, clean & replace.

If you see something like a wiring harness coming out of the speedometer cable (how it's been described to me) then you might have a loose connector in the cluster, though this would make it run & shift poorly. That part is beyond my anecdotal knowledge.

Disclaimer: I haven't worked on either version.

Are the replacement engine/trans the same as original or a year/performance swap? I've heard of some combinations that cross years that required extra work due to sensor and computer changes.

The car is a 68, and the engine and tranny came out of a 69.
It runs and shifts fine, except for the horrible vibrating problem.

Ah, ignore everything I wrote before.

It'll have a standard old mechanical cable (~1cm across) coming out of the tail of the transmission housing. Pointing ~45* down & toward the driver's side? People often neglect to connect it, or to resolve gear size issues at the end of a swap.

Yup just a good ol standard style cable is what it is...should hook up near the tailshaft of the tranny and straight into the back of the gauge, if the speedo still doesn't work, take a short cruise with the speedo disconnected and make sure the cable is turning.

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.