Luckily this is apparently a common problem with the iPhone 4S, so the genius at the Apple store (and I use the term with disrespect for Apple's marketing department, not the guy in question, who was perfectly nice) knew exactly how to fix it: either he could sell me a new iPhone 4S today for $199, or I could wait until June, when my service provider would sell me a new iPhone 5 for $199.
This has been said many times before, but still: we're gross. My grandfather has a business repairing printing presses, many of which are older than he is, and all of which are based on a technology from the 15th century. My generation, by contrast, fetishizes devices which are effectively designed to be discarded on a one- or two-year cycle.
In any case, here's the workaround, which I intend to use until the whole phone dies and I switch back to bottles with letters rolled up inside, etc: if you enable assistive touch, it creates a button on your screen that lets you lock or turn off the phone without using the power button. This is how you enable assistive touch.
As an amazing aside, some of you know that my brother Mark is intermittently spooky. Yesterday, thinking that maybe a piece of dust had gotten stuck under the power button, I was wandering around the house asking if anyone had a can of compressed air. This is the only time in my life that I've asked anyone that question. An hour later, we were opening gifts. This is what Mark had inexplicably but entirely correctly thought would be a great Christmas gift: