If I understand your question right, a function that truncates your text after stripping HTML is what you need.
What about using wp_trim_words():
<?php while($breakingnews->have_posts()) : $breakingnews->the_post(); ?>
<a href="https://wordpress.stackexchange.com/questions/253344/<?php the_permalink(); ?>"><?php the_post_thumbnail('post-image'); ?></a>
<div class="leadnewsboxtitle">
<h2><a href="https://wordpress.stackexchange.com/questions/253344/<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>
<p><?php echo wp_trim_words( get_the_content(), 60, '...' ); ?></p>
</div>
<?php endwhile; ?>
General usage:
wp_trim_words( string $text, int $num_words = 55, string $more = null )