Stephen,
I just tried on your site, and the invite was properly sent to my notifications area in Facebook (see image below). However, the Invite module itself was cut-off on your site, so that may be part of the problem. Basically, our module creates an iFrame to Facebook and sets up the invitation window. Unfortunately, Facebook resizes this window as you select friends, click "Send Invites" and go through the process. On your site, after I selected a friend, the first window came up to confirm the sending of Invites. Then, another window came up to send invites by email. This window was taller than the module area you put JFBCInvite in, and the "Skip" button was missing at the bottom.
I still managed to hit the Skip button, knowing it was there, but that may be part of your issue. Unfortunately, there's not much we can do about this in the JFBCInvite module, because it seems like the slider thing isn't detecting the larger height. Can you try the Invite module on it's own page, just to make sure the invites are actually being sent? Then, if so, we can swing back around to try and help you getting the slider module working better.. however, that may require you contacting the developers of that module.
Hope this helps get you going!
Alex