Doesn't look like there's a way to single out the Desktop via the 'Manage Storage' GUI.
However, you can always just symlink any directory from your iCloud drive; that way you can ensure that only the Desktop folder is being synced. This is with the caveat that while your Desktop files are being stored in iCloud, they're still taking up space on your HD before, during, and after sync. If you do it on every Mac then you'll keep your Desktop in sync between them all (files added, deleted, changed etc); this might not be what you're after though.
If symlinking isn't something you know how to do, instructions are below.
Backup (via TimeMachine or whatever you use).
Then fire up Terminal and run the following to move your Desktop folder into your iCloud Drive:
mv ~/Desktop /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/
Then just symlink it to the usual 'Desktop' location:
ln -s /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/Desktop /Users/<yourusername>/Desktop
Now anything that you add to ~/Desktop (by dropping things onto your desktop etc) will actually be dropped into your iCloud drive. They are of course still taking up local space on your machine. When you delete something from your desktop then it's gone from the 'Cloud' as well. The benefits of this are more for sync between machines (+ access via your iPhone of course).
If you've got another mac, move anything you value from it's desktop folder into your iCloud Drive 'Desktop' folder first, then delete ~/Desktop on that machine and symlink again as above.
Best Answer
I had to leave this as an answer, because it's too long to explain in the comments section. I'm not guaranteeing that your invisible 13GB of files are definitely hanging around in the iCloud trash, but it's just the most obvious first guess that I could think of. So, let's find out:
Open up Terminal and type in this command:
That will take you to the local storage folder where iCloud keeps copies of your files on your computer. Type
ls
to list the files in the directory if you're curious, but it's much the same as navigating to your iCloud folder in Finder.However, what you actually want to do is see if the iCloud trash is still full of files that are taking up space:
Hopefully, it'll contain about 13GB of files in it. If this is the case, and you're sure there's nothing in the trash folder you wish to keep, just delete the folder:
This is permanent and cannot be undone. So be sure you're happy to lose those files in the trash.
Then type
exit
and quit Terminal.