Sorry for the delay. The problem isn't the 1000px div. That's something Facebook does inside their iFrame and is normal (though it looks funny). The problem is that the div you're placing that code in is thinking the login button is 100% width.
If you add the following to your stylesheet, it should fix things up for you:
#ja-search .jfbclogin {
width: 100px;
}
Everything else seems right in the HTML and the Facebook authentication popup properly appears. I didn't go farther than that though. If you run into more issues though, just let us know.
Thanks,
Alex