I think this is because you are using client software and not the server.
mysql is client mysqld is the serverTry:
sudo service mysqld start
To check that service is running use: ps -ef | grep mysql | grep -v grep.
Uninstalling:
sudo apt-get purge mysql-server
sudo apt-get autoremove
sudo apt-get autoclean
Re-Installing:
sudo apt-get update
sudo apt-get install mysql-server
Backup entire folder before doing this:
sudo rm /etc/apt/apt.conf.d/50unattended-upgrades*
sudo apt-get update
sudo apt-get upgrade