it was hectical here and I obviously overlooked that I didn't answer your question at all. My apologies for the now very late answer.
It is not possible to get what you want with my plugin, since the custom login page only styles the wp-login.php after your wishes. For your goals you would need the function
wp_login_form()of which you find the documentation here.
Or you will need a plugin that gives you a widget with a login form for a widget area in the footer. The only thing that you won't have in your form is your logo. But perhaps you'll find a plugin that provides you with a space for the logo on top of the function as well.