How do I retrieve the category ID (ugly permalinks) in my sub-navigation menu?

Use get_term_link with the slug, and specify the taxonomy:

$cat_slug = 'Amazing';
$taxonomy = 'category';
$cat_link = get_term_link( $cat_slug, $taxonomy );

if ( is_wp_error( $cat_link ) ) {
   echo $cat_link->get_error_message();
} else {
    echo $cat_link;
}