Ivan, thank you for that. The power supply failure sounds like good advice as something to pursue (I'm thinking of writing all this up afterwards as a guide for someone in my situation). I think however that, since my last post, I've got to a point where I can rule that out without specifically checking for it.
Following the advice elsewhere on this forum, I installed a third-party application that would allow my Win10 box to read Linux-format disks - not Ext2IFS, which crashed the PC on installation, but DiskInternals Linux Reader, which seems pretty good. And I went out and bought a docking station.
The result: one drive is readable in DiskInternals Linux Reader, one isn't.
The left of the two drives as you look at the front of the DNS is the readable one. I'm guessing that's HDD1, which is the one that failed second. DiskInternals Linux Reader gives an error message saying it's damaged but repairable.
The other shows up as a drive but can't be read. It also has some mottling to its casing over the disk itself - could that be heat damage? I have no idea, but the other one has only a small amount of it.
The drive that can be read seems completely intact - I can drill down to individual files and so far haven't found any part of the disk I can't access. So I'm very hopeful that I can copy its contents to another drive and then thank the stars for my lucky escape. I'm not taking anything for granted yet, but at least I know everything seems to be there.
Still some work to do, but it looks encouraging.
Oh, and the other back-up drive that I thought hadn't run for a year? March, as it turns out. Could have been worse.
Thank you both - I need to get on with fixing this now! Will report back on progress...
EDIT: I see from another thread here that HDD2 is the left drive. That kind of makes sense in this context, I guess. Anyway, after an 11 hour transfer it looks like I have managed to copy the whole contents of the drive (more than 350,000 files) onto back-up storage.