So I turned my wifes PC off and on again after staring at it for the requisite amount of time and realising it wasn't going to fix itself and all the things I'd just tried hadn't worked.
Fine after a first reboot.
Couple of days later same thing on my own machine. My machine is a bit more annoying as it's a floor down from the router so not as easy to keep an eye on whether connectivity has returned or not. Again, after checking all the cables were fine I just ended up rebooting and was fine.
The previous article to this says that rebooting doesn't always work but my experience from this was that it did the trick on those two machines.
As for "how am I meant to patch this when I can't get online?" - this is no different to those times when your network card driver would go kaput and you'd need to download a new / updated one. The good old days were just as bad and it hasn't always been Microsoft's fault.