I am not that expert on WWW::Mechanize but lately I have been trying do something similar to this, and so far I haven't had much luck with it. But looks to me your issue is JavaScript. If you take closer look at the "loginform" they call a quicklogin() function. WWW::Mechanize doesn't deal with javascript. Search for LWP on cpan and try LWP->agent.
Your problem is right here... (at least the way I think...)
<form method="post" name="loginform" action="https://login.facebook.co
+m/login.php" onsubmit="quicklogin();">