As you have seen the message /email focuses on advising you of a failed login though never produces a code.
To get around this, go to your account settings
https://accounts.logme.in/accountsettings.aspx
Then under notifications, untick the option for failed login attempts.
Save/apply
Then go attempt to login again, and it will notify of the code being sent
Then check your mail, it should now appear
It seems that notification is take priority the code generation email
Poor design really. Cannot see how they overlooked this