How can get the last post date of the user?

In your sample code the $latest_posts is an instance of WP_Query class.
If you want to get the latest post’s date, you should do this:

$latest_posts = new WP_Query( $args );

$last_date="";
if ( $latest_posts->have_posts() ) {
   $latest_posts->the_post();
   $last_date = get_the_date();
}

return $last_date;