Yup, that's tacky

There's quite a few options available, depending on how easier vs customizeable you want it to be.
Including with JFBConnect and JLinked is our SCLogin module (v2.0) which is a basic replacement for the Joomla login module but has the social buttons added to it. This would 'work', but not well with your template.
There is also a 'new' SCLogin module which will be including with the next release of JFBConnect & JLinked. You can see it in action at the top of this page, but there's lots of different styles if you don't want a popup. I think this is your best option specifically because it has a
'horizontal' mode option
that I think will hopefully work well in the area of your current login area at the top. For the full product overview or to download the SCLogin module (it's free), see:
www.sourcecoast.com/sclogin/features/
In addition to that, there are other options, such as the {JFBCLogin} and {JLinkedLogin} tag that you can put anywhere on your site to have those buttons appear. These are good in an article or Custom HTML module to give maximum flexibility. You can even use your own custom images for the buttons if you want.
So, there's a lot of ways to add the social login buttons, and I hope one of the above sounds good. If you have any questions though, just let me know.
Finally, for why you can't login with Facebook, when I try, there's the following Javascript error that comes up in the browser console:
Given URL is not allowed by the Application configuration.: One or more of the given URLs is not allowed by the App's settings. It must match the Website URL or Canvas URL, or the domain must be a subdomain of one of the App's domains.
That means that Autotune hasn't set the right values for your page. Can you please re-run autotune again and hit 'refresh'. Make sure that the Site URL has 'http://www.
.com' and that all other recommendations are right. There's nothing else I can see that would be a problem for Facebook login.
I hope that helps,
Alex