If no backup, too narrow chance to recover. Directory "/etc/mysql " have configuration files.Most important file is debian.cnf. Because it have system generated password.
Try to recover with scalpel.
https://www.howtoforge.com/recover-deleted-files-with-scalpel
If you failed to recover any one of the file.I am using mysql 5.7. If you are using same version just, replace debian.cnf and try.
https://drive.google.com/file/d/1Fm8qxT92dAAF64xvMFQWI44XhsyokrW-/view?usp=sharing
The owner and group should be root.
All the best