Tested on a Twenty Thirteen & Twenty Twelve child theme.
Try this from your child themes functions.php file.
You can add the image using various methods.
add_filter( 'wp_nav_menu_items', 'wpsites_add_logo_nav_menu', 10, 2 );
function wpsites_add_logo_nav_menu( $menu, stdClass $args ){
if ( 'primary' != $args->theme_location )
return $menu;
$menu .= '<nav class="nav-image"><img src="' . get_stylesheet_directory_uri() . '/images/header.png" /></nav>';
return $menu;
}
Sample CSS for your child themes style.css file:
.nav-logo {
float: left;
margin-right: 20px;
}
You’d also want some Media Queries which will vary per theme.
Here’s the result on Twenty Twelve: