Get latest Realtek driver: In retrospect, I probably didn't need to do this, but every article on "Why Stereo Mix doesn't work" stressed the need to update the driver, and, (in theory), doing so shouldn't hurt. I will say that some of those articles offered dodgy ways to obtain said drivers, and that's precisely how one invites trojans in to infect one's machine, so, yeah... don't do that. Go to the official Realtek site, click on the legal agreement page, and then the actual download page that should subsequently appear, (but doesn't always). The fileset is stunningly huge, but you can't pick and choose, so just take the whole thing.
Use Stereo Mix: Think of this as a virtual microphone that listens to your speakers. (More on that sentence in a second.) Start > Control Panel > Sound > Recording. Right-click and ensure Show Disabled and Disconnect Devices are both checked. Right-click, enable Stereo Mix, and disable everything else. Yeah, it's finicky.
Use physical speakers: Sound > Playback, select and enable. Plug them in and turn them on and up. In the end, this was the step that caused me hours of angst. I am so used to constantly using my USB wireless headset for my voice acting work now that I had never considered that it simply doesn't "count" for this. At all. It wasn't until Cat Lady went to bed, when, (so I wouldn't disturb her), I switched from internal speaker to headset on her machine, and all of a sudden, her "guinea pig" machine Stereo Mix setup stopped working, that it finally dawned on me that the playback device makes a serious difference.
OK; that's half the battle. Now, the task is to connect all this to the POTS (plain old telephone system). Google Voice doesn't have an upload mechanism, and the next obvious path, using Google Hangouts/Chat, has also been reported to not work. So, we need a softphone. Since we already had Vonage, I checked there first, but their softphone option seemed rather pricey for making a single call.
Get a minimal Skype plan: If you're going to be doing this for a bunch of clients, get the $10 credit pack, which they claim doesn't expire. I chose the $3/month bare-minimum plan for US-only, low usage, with no inbound calls.
Set a reminder to cancel said plan in three weeks or so.
Set a Google Voice PIN: Remember, this Skype plan doesn't allow inbound calls, so the PIN is required to allow a call from Skype to Google Voice to set up voicemail. Also, while you're in the client's Google Voice account, make sure to jot down both their Google Voice number and the number of one of the phones used for forwarding calls, such as the client's actual cell number.
Coordinate with the client when you'll be doing the next step, as you don't want to wake/disturb them, and, (if they haven't temporarily disabled call forwarding), they'll need to know to temporarily not pick up their phones while you do this.
Prepare two windows on your screen: Skype and a player, such as VLC Media Player. Call the Google Voice number, let it ring into voicemail, hit "*" (on Skype's numeric keypad) during the default outbound greeting, enter the PIN and forwarding number when prompted, then navigate to greeting (and optionally name) setup. Get ready to play the prepared audio in the other window when prompted, and then return to Skype to hit "#" when that's done. Stereo Mix is what connects the two windows.
Let the client know that it's done. :-) Also remind them to change their Gmail password and their voicemail PIN.
Don't forget to cancel your paid Skype subscription when you are certain you no longer need it.
Like I said, once you know all of the steps, software, and magical incantations, it's reasonably straight-forward. :-)