Your includes/modules/pages/contact_us/header_php.php looks correct. Just be sure the rest of the files in this plugin have been loaded into the correct locations.

While like unrelated, you have a mixed content problem on your site.

<script src="http://cdnjs.cloudflare.com/ajax/libs/jquery-easing/1.3/jquery.easing.min.js" type="text/javascript"></script>

is being callled insecurely so is not loading. Fix that and then check the CAPTCHA again.