Thanks Ross, unfortunately this site is internal. But I did do a console view from inspect element and it was showing that the pages were trying to load http files instead of https. I went to the theme and found it was loading the location thus:
<link rel="stylesheet" type="text/css" href="<?php bloginfo('stylesheet_url'); ?>" media="screen" />
<link rel="stylesheet" type="text/css" media="all" href="<?php bloginfo('template_directory'); ?>/css/effects.css" />
Our IT department did a roll back so we are back to 3.4.2 and everything is working again. The theme has always been loading the files this way so I'm not sure why going to 3.5 changed anything. I unfortunately don't have access to the db so I can't look to see where all the stored http values are, which is why I've been using this plugin :)
I've checked all sorts of settings like making sure home and site url specify the https but it wouldn't take and I ended up in a loop when I tried to force admin ssl through the wp-config.php file.
Since we are back to the old version I can't really look into this more. But thanks for your speedy reply!