The sourdough is alive once more.
I don't know what exactly happened before -- I suspect I wasn't mixing the starter up enough when I measured it out for baking -- but I had a problem for a few weeks with my sourdough starter where it just wasn't working right. It would take over a day sometimes for a batch of dough to rise, and it didn't taste right any more. The bread was too dense, the crust was too flat, the bagels too hard and floury.
I finally gave up two weeks ago and bought some conventional yeast packets at the store. I mixed one of them in with my inert starter, and watched it froth back to life in no time. The frothing has continued through a pizza crust, two loaves of bread, some hamburger buns, and now a batch of bagels. The sourdough is alive and, what's more, it tastes like sourdough again.