Thanks for all the responses. I refused to let the shifter beat me, so I spent a good amount of time figuring it out until I got the technique right. I have a Falcon MLG-26i twist shifter, which is meant to replace my broken SRAM MRX 200-60. The mechanism, layout and installation is identical for both. Perhaps there is a way of installing it without opening the shifter, but I haven't figured that out.
I actually made a video of it so that I would remember next time I need to install another grip shift cable. My camera is not very clear, but if anyone ever needs the video, let me know and I can email it or something. It's 17MB in size, otherwise I'd post it somewhere.