Adding body class to html tag that already has language attrubutes?

Maybe you could use a filter to add the current language of the website to the body class.

add_filter('body_class', 'language_in_body_classes');
function language_in_body_classes($classes) {

    $classes[] = get_bloginfo('language');

    return $classes;
}

Edit : maybe I misunderstood what you wanted. There is actually a language_attributes filter but I do not think it would be very standard to add classes to the html language attr.