You can use getimagesize()
for this purpose:
list($width, $height) = getimagesize(esc_url( get_theme_mod( 'mitch_logo' ) ));
echo ".logo {
width:{$width}px;
height:{$height}px;
background: ...;
}";
If that adds a little milliseconds to your page load time, I can edit this to provide a JavaScript version of the answer 🙂
Hope that helps.