What do I need to do to convert my MultiSite from HTTP to HTTPS?

You could run a script to UPDATE all urls and guids to https, if you want a clean setup.

But also consider alternatives such as:

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteCond %{SERVER_PORT} 80 
RewriteRule ^(.*)$ https://www.yoursite.com/$1 [R,L]
</IfModule>

In wp-config.php for the backend:

define('FORCE_SSL_ADMIN', true);

In wp-config.php for the frontend (or run a db UPDATE script):

define('WP_HOME','https://example.com');
define('WP_SITEURL','https://example.com');

Then you could simply run a script to UPDATE all wp_posts content url.