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The speedo has a magnetic component in its drive, the odometer is based on plastic gears. When they loosen on the shaft they no longer turn and in some cases the gear can strip. That is why both speedo and odometer can be calibrated separately.
If the worm gear referenced in the DIY is loose, the shaft will move out from the gears it's supposed to drive and your odo will be non-op. The basic problem is that the gray number wheel on the end of the axle is supposed to be tight to the little axle. If it's not, it will slip and not drive any of the odo number wheels. You can, with patience, slide out the axle just enough to give it a "nice" tap with ball pein hammer, the object being to deform the axle just enough so that when you push the gear back into place, it binds on the shaft.
I've done it a couple times in 2002s and my CS speedo. You can also, if you have more patience, reset all the number wheels to zero. If done for the purposes of defrauding a customer in a sale, this is of course illegal and most states have laws stating what you can and cannot do to an odometer. I bought my CS with a non-op odometer and left it alone for over 15 or so years, before I got the gumption to remove the speedo, fix the clock (a whole 'nother story) clean the speedo and clock faces, and blip the odo shaft so it works.
It isn't "a" rainy day project, best you allot several days so your immediate level of patience isn't exceeded. But OTOH it's "one of those things" that most coupe owners will do at one point or another. Some use checkbooks to make it happen, but professionals for this sort of thing need to be state licensed (at least here in Hawaii).
Got it out and disassembled. I sat there for about 30 minutes playing with it, spinning the speedo cable input with a small Allen wrench to assess the issue. Turns out the worm screw was not engaging with the worm gear; circled on the left in the photo below. I loosened the screw in the circle on the right, repositioned the worm screw to engage, and tightened it up. I then applied a tiny bit of grease to each of the gears, verified operation, and applied a dab of paint to the screw.
This car was grey market from 1985 (originally from The Netherlands) and the back of the speedo housing is etched with 10/85 MPH to show it was converted as part of the federalization.
The paint is there to serve in lieu of "Lock-Tite", so it doesn't loosen (hah!) over time and cause exactly what the OP encountered. The paint spot is encountered on other low-torque, relatively delicate gizmos, that can't take a torque value, not that anyone has a 1/8" torque wrench for those tiny screws.
BTW to the OP: you got lucky, once the speedo was out, this was the simplest of all things that could have gone wrong on the speedo. Back in and away you go, or reset the mileage and you'll have a new CS!