At first, I thought this was completely broken, but then, by accident, I discovered that it would work if I did one simple thing. After placing the widget where I wanted it in Appearances > Widgets, I refreshed the page, and it started working. I was able to enter content, use the TinyMCE buttons, and save my changes.
The one remaining problem is that I also use Ultimate TinyMCE, so I have additional buttons that this cuts off. I had to go into the Ultimate TinyMCE controls and drop those buttons down to a third line so that they'd show in this. It would be nice if the widget box could be sized horizontally, but at least I got it working.