Hi Mike,
I see this as a totally mechanical repair. Looking at the pics, if this was mine, once I found a worm gear with the correct thread size, not necessarily diameter at this point, though that would also be great, I'd drill it out to slip onto the existing drive shaft if there is no hole already. Don't glue anything yet, slip it on the shaft to check and adjust length etc...until satisfied with the test fit. Then sand down the worm gear teeth to the proper diameter to sit in to the volume shaft gear. I'd do this with the gear on a drill to spin it as I sand it down to the correct diameter to sit in the volume gear properly. Let's hope the gears you bought have one, or a couple that fit in case you need to try it a couple times. After you are satisfied with the fit, use JB Weld to glue the gear to the existing remaining gear shaft. Obviously you'll need to clean off all the old grease for the JB Weld to hold.
Glenn