Hi KarlaCluft,
I would try the $mechanize->submit_form(); function.
Some thing like this maybe:

$mech->get($url);
$mech->submit_form(
form_name => 'signin_form',
fields => { 'Username' => $user,
            'Password' => $pass}
);
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If that doesn't work for you, do what runrig said. Also, I recently did a brief tutorial on WWW::Mechanize in our tutorials section if you want to take a look at that.
UPDATE: With your current code, you could also try this instead of just $mech->click();

$mech->click_button( value=> "Login");
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If that button is the one needed to submit the user and login and if submit_form doesn't work..