I figured that if one drive had gone, the other drive was now a ticking time bomb, so I copied the entire musical contents of the drive (something on the close order of 300 MB) off to the NAS drive upstairs. Then I went on Amazon and ordered a new pair of hard drives with "same day delivery", which in this case meant that I would get them on Tuesday. Fortunately, this only cost $5.99 on top of Amazon Prime, so it was actually a much better deal than trying to get them from most anywhere else.
The pair arrived this afternoon and I strapped one of them in to replace the failed drive. The rebuild of the mirrored pair is now up to 97% complete, so it should finish by morning. Then I can strap in the other new drive to replace the working (but possibly ticking time bomb) drive and let it rebuild the RAID mirror again.
And then I can get back to it. :)