Firefox 58+ versions have switched to key4.db (SQLite) and no longer uses key3.db (DBM) for the key file that stores the encryption key for logins.json, but can fallback to key3.db if you remove key4.db. It is possible that something went wrong with the migration from key3.db to key4.db. If you have key3.db in the profile folder then remove the current key4.db file to see if the migrations works better this time. If you have logins.json.corrupt then rename this file to logins.json. Make sure to keep a backup of all involved files (logins.json and key3.db and key4.db). You need to make sure Firefox is closed when you make modifications to files in the profile folder.

You can use the button on the "Help -> Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page.

• Help -> Troubleshooting Information -> Profile Directory:
  Windows: Show Folder; Linux: Open Directory; Mac: Show in Finder
• http://kb.mozillazine.org/Profile_folder_-_Firefox

( LOOKS LIKE I WAS TOO LATE, ANSWER ABOVE IS BETTER D; ) You will also need a file called "key#.db" where # is a numeric character (such as 4 in my case) so for example, I have logins.json and key4.db. If your logins.json.corrupt is in proper format, it should be ok to rename it to logins.json. And therefore so long as you have the key#.db, firefox will be able to unencrypt the logins.json passwords.

If you're missing the key#.db file you're pretty much screwed unless you have a way to find a backup/old version of that file.