I entered my email and password and tapped the login button. Yet I still can't log in.

If there is a space in your Kakao Account email or password, the login button will not be activated (i.e. turn yellow).
Make sure there is no space in your Kakao Account email or password, and your password is at least 4 characters.

If your password includes special characters, change your keyboard setting to the default setting provided by your OS and try again.