I see the following javascript error on your page: "Invalid App Id: Must be a number or numeric string representing the application id." You will need to resolve this before the button will work.
Additionally, we have a new method for adding custom login buttons, which is described here
www.sourcecoast.com/jfbconnect/docs/social-login/login-button - this might help you out as well. You can add the file to the directory structure as listed and then use the JLinkedLogin easy-tag to add it to the page. For more information on the easy-tag, see JFBConnect > Social > Examples in your site backend for the LinkedIn widgets.
-Melissa