My sympathies! In hindsight, I would have saved so much time and bruises if I had just gone straight for the dremel. It was very difficult, navigating the tight space under the sink and trying to get leverage in the tiny gap behind the basins. Wasted so much time (hours!) trying to get that nut off of there "nicely" when I had been intending to throw away the old faucet anyway. I called in my roommate Fishy to help me, and after only a couple of tries at the nut, he quickly suggested cutting it off with the dremel. I had to pause and think "oh yeah, I'm not planning to save this old faucet, so that might actually work". Once I'd gotten past that cognitive barrier, the old faucet came off in, literally, seconds.

In other news, I'm having fun experimenting with different resistor values in the voltage divider. Still, even when I have the ratio of the resistors set so that the output voltage is something like 1% of input voltage, it's still coming out too loud. Like, way louder than the unmodified headphone jack output that I was originally using. I believe the results I'm getting are usable for my purposes at the moment, though not ideal.