I already said I didn't want to do a repair install, a.k.a. an in-place upgrade. So I went ahead and tried a few more things first on the off chance they'd help. I reset the TCP/IP stack, reset the Winsock catalog, and I blew away the Software Distribution folder and let all of the Windows updates be detected over again, etc. It had been on a domain too, so I went through local Group Policy and undid all of the settings that had been applied from the domain Group Policy. (I also forgot to mention in my first post that I'd run a CHKDSK /R too and no issues were found.) None of those things made a difference - KB2871997 still wouldn't install.
Since it didn't seem like there would be any more useful information forthcoming here, I decided I might have to do a repair install after all. I wanted to avoid doing that because I'd found that while my customer had my machine they hadn't simply put the PC on their domain and installed their corporate McAfee on it, they actually imaged the entire machine, and I wasn't sure what media/installer they'd used: OEM (the PC is a Dell), retail, or volume, etc. After pulling up some info using SLMGR and a couple of third-party tools, I actually found that they installed/activated Windows using a Key Management Server (KMS) which requires the PC to hit the KMS periodically to renew its activation. Well on this machine that wasn't going to work at all going forward. So I'm about to reinstall the entire OS from scratch or put one of my images on it.
So I'm not happy that I spent so much time on this. We often loan PCs to customers or prospective customers as demos, but they shouldn't have imaged the computer - especially without informing me. It's possible that something in their image or something related to one of the applications they installed caused this problem. I would've reinstalled it right when it came back if I knew that.
However, Microsoft still needs to address this. I doubt that the other people posting in this thread are having the problem on a PC that was subject to the same conditions as mine was. So Microsoft still needs to "fix" this update or at least provide some more detailed or useful information about what it's for and what things might cause the update to fail.