Create 2 pages category-news.php & category-video.php & put this code in them. Then customize the markup for both as you like
<?php
$children = get_categories('child_of'=>get_query_var('cat'));
$cat = array(get_query_var('cat'));
foreach($children as $child)
$cat[] = $child->term_id;
$catPosts = new WP_Query( array( 'category__in' => $cat, 'posts_per_page' => 5, 'orderby' => 'date', 'order' => 'DESC' ) );
while ($catPosts->have_posts()) : $catPosts->the_post(); ?>
<h1><a href="https://wordpress.stackexchange.com/questions/24423/<?php the_permalink() ?>"><?php the_title(); ?></a></h1>
<div class="meta">
By <?php the_author() ?>
</div>
<div class="content">
<?php the_excerpt(); ?>
</div>