URLs not being output with https

Is the site running on Apache? If so you could put this in your .htaccess file

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

also if you need SSL in your admin area and login pages put this into wp-config.php

define('FORCE_SSL_ADMIN', true);

here’s another way that is more generic:

RewriteEngine On
RewriteCond %{HTTPS} !=on
RewriteRule ^ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]