First Aid shows "File system check exit code is 8." and does not erase the disk on time machine

The time machine is sitting in another room, separated from my iMac via two switches, if that is relevant.

Some clarification would help. Time Machine is software, so saying it's sitting in another room doesn't make any sense to me. Apple's Time Capsule is hardware in a white plastic enclosure, and erasing its internal hard disk drive requires using AirPort Utility, not Disk Utility. The presence of one or more switches is OK.

Also, you are using Disk Utility to repair a sparse bundle disk image. Why? Next, that disk image is formatted as case-sensitive and not journaled, whereas a Time Machine backup device should be journaled. Case sensitivity is optional in HFS+ and should generally be avoided unless the source volume is also case sensitive.

Although it is likely the Time Machine backup device has failed, more information is required.

i've had the same problem a number of times. When first aid starts it unmounts the disk and when it encounters an error it quits without mounting it again. Then I can't do anything with disk utility on that disk and the option to mount the disk does not work.

Quit disk utility and start it again. It may take a minute or two to find and mount the disk. Then you can erase it or run first aid again. Sometimes running first aid a second time finds no errors.

It appears the drive may have failed. If possible, attach the drive to the computer, and try First Aid/Erase again. That will eliminate any possible network interference.