Well, my misconceptions about how the bottom end of the cable is constructed has cost me the price of the assembly; if I knew a little more I could have gotten away with replacing only the inner cable.
I had inserted the 7/8 piece of cable and pressed on the end of it, causing the other broken end of it to press against the matching broken end of the 1/8 piece of cable, then turned the slot at the bottom end with a screwdriver. Because the bottom end didn't come out, I had assumed that the bottom end of the cable had a square cross-section like the top end and inserted into a square hole in the end of an adapter (the other end of which I could put my screwdriver into) that was captive in the bottom end of the cable. This is not the case. The "adapter" is actually crimped onto the end of the cable, and If I had pushed harder on the top end of the 7/8 piece, I might have popped the "adapter" out of the bottom end of the cable sleeve. It turns out the two fingers that form the bottom driven end of the cable had splayed apart slightly, wearing a slight ridge inside the bottom end of the sleeve, preventing the cable from easily falling out the bottom end of the sleeve.
I found this out by hacksawing the sleeve apart near the break, then pulling it apart. I was expecting to be able to pull on the 1/8 piece broken cable end, pulling it out of the cut sleeve end. I was surprised when this didn't work. Pushing hard on the broken end finally allowed the 1/8 piece to pop out the bottom end of the sleeve.
My kids' school fees are due, so I'll be riding around without a working speedo until my next paycheque.
Spayed fingers:
Evidence of grinding:
