To get Time Machine going again, you need to remove the errant files with the following steps:
Ensure your backup device is turned on and connected to your computer (wirelessly or wired, it doesn't matter)
Turn off Time Machine using the big button in Time Machine preferences. NOTE: I did not execute this step and was still able to fix Time Machine.
Make sure the backup volume (which is just part of the backup drive) is mounted to your desktop/Finder. You may have to initiate (and subsequently cancel) a Time Machine backup process in order to see the drive on the desktop or in Finder.
Access your backup volume in Finder (it should be named "Backup of your-computer-name..." or something similar) and double-click into the "Backups.backupdb" folder.
Next, click into the "your-computer-name" folder. You should see a bunch of folders with dates. These are all the dates corresponding to days you ran a successful backup.
At the bottom of the folder listing will be one or two things you need to move to the trash. You may see only one or both of these, but delete the file that starts with a date (it should be the date the backup failure started) and ends in ".inProgress." Also delete a file named "Latest" if it exists.
Return to Time Machine preferences and turn Time Machine on. Remember, I was successful without performing this step.
Initiate a backup using the Time Machine drop-down in the menu bar or wait until the next scheduled backup run.
A final but important step, be patient. Time Machine may sit in "preparing" mode for a while. It's important to at least let it run for an hour or so to see if the process will continue actually writing files to your backup volume.
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