My time machine backup disk suddenly isn't being mounted. fsck_hfs -fy -x fails with exit code 8

My time machine backup disk is connected via USB to my MacBook Air and after I rebooted today, the light on the disk is flashing, as if it's doing something, but is isn't being automatically mounted, as it normally would be. The disk has been sitting on my desk, next to the Mac, for a few days without being moved.

I started Disk Utility and ran First Aid with these results:

First Aid on the physical disk came back with Operation Successful.

Then First Aid on the logical volume under the physical disk:

I have run First Aid on the logical volume four times, each time with the same result as above. I have internet searched for error code -69845 but couldn't find anything useful. And a search for error code 8 isn't helpful either - just that it's a file system error.

The disk has the last six months of Time Machine backups on it so I'd rather not lose it all. And I now don't have a working backup. Any ideas on how to fix this?