I agree with all the above. You should be able to access the tailshaft by following the procedures Randall and John posted but I don't think you will need to drain the cooling system. With the tranny supported and angled just slightly up at the rear you should be able to take the rear housing off the geabox to access the worm gear. If your worm gear has a metal hub, you will need a puller to remove it like John said.
You should be able to visually inspect the worm gear through the hole where the pinion gear was removed. Even if you cannot see it you should be able to reach in the hole with a probe or pick and find the teeth on the worm. You should NOT be able to spin the worm or slide it forward/aft on the tailshaft. If you can spin it easily there is a problem with the gear.
My Mini has a metric speedometer and while it is easy to multiply kilometers by 0.6 (round number) to get MPH, I tinkered with a few alternate speedometers. You can use a Sigma (brand) bicycle computer to have speedometer, odometer, and trip functions. However, the bike computers are battery powered and don't have a good solution for night illumination. It's not practical to push the "light" button so you can see the display for 15 seconds or so. Later I bought a portable, digital GPS speedo through eBay, $30 delivered. I believe the brand I bought was "LTC" or something similar. It is about the size of a pack of cigarettes but 1/2 the thickness. There is not a true odometer or trip function but if you just want speed, it's hard to beat. All you do is plug-and-go. However, you do sometimes have to wait about 3 minutes while satellites are acquired before the speedometer starts working.