Which are the MediaElement.js scripts to enqueue

I had the same issue, but with the audio player.

First of all, you need enqueue MediaElement styles

wp_enqueue_style( 'wp-mediaelement' );

Next you need to add wp-mediaelement as dependency of your app script or enqueue it.

wp_enqueue_script('wp-mediaelement');

or

wp_enqueue_script( 'app', get_stylesheet_directory_uri() . '/js/app.js', array( 'jquery', 'wp-mediaelement' ));

As result you need something like that:

function my_enqueue_scripts(){
    wp_enqueue_style( 'wp-mediaelement' );
    wp_enqueue_script( 'app', get_stylesheet_directory_uri() . '/js/app.js', array( 'jquery', 'wp-mediaelement' ));
}

add_action( 'wp_enqueue_scripts', 'my_enqueue_scripts' );