Disk Utility cannot repair my external Time Machine drive.

You cannot, you need to replace the External Hard Disk and start over. As a general rule of thumb it is wise to keep a HD for about 4-6 years and then replace it as the chances for failure increase. Also, it is wise to buy a high quality elcosure so the drive gets adequate cooling. The enclosures Walmart style EHDs use are pretty poor and get lousy cooling. These enclosures include WD, Toshiba and Seagate. Many experienced users spend a few more dollars and buy high quality enclosuresto increase cooling and extend the life of the drive. Personally I prefer the OWC Mercury Elite Pro. One of the high points of these is when the HD dies it can be replaced in less than 5 minutes with a simple screw driver, which means the enclosure is not throw away ending up in some landfill. I have some that are 15 years old and still work fine.

Following my underwear rule: "If its doubtful, its dirty", I would get a new external and start a new TIme machine backup. Then reformat this disk and use it as a second belt and suspenders backup or set it up as a bootable clone.