1) You built a POST request, but didn't actually send it.
2) I'd suggest using WWW::Mechanize instead of hand rolled UA calls, let it handle the nitty-gritty.

#!/usr/bin/env perl

use strict;
use HTTP::Cookies;
use WWW::Mechanize;
use Data::Dumper;

my ($username, $password);
while (!$username || !$password) {
    ### Prompt for username
    print "\nAkamai Administrator credentials\n";
    print "username> ";
    $username = <STDIN>;

    ### Prompt for password
    print "password> ";
    system 'stty -echo';
    $password = <STDIN>;
    system 'stty echo';
    chomp($username, $password);
}
my $login_url = 'https://control.akamai.com/EdgeAuth/login.jsp';
my $real_page = 'https://control.akamai.com/home/view/main';

my $mech = WWW::Mechanize->new();
$mech->agent("Mozilla/5.0 (Macintosh; Intel Mac OS X 10.7; rv:12.0) Ge
+cko/20100101 Firefox/12.0");
$mech->get( $real_page );
if ( $mech->base =~ m/$login_url/)
{
    $mech->submit_form(
        form_number => 2,
        fields => {
            "User ID" => $username,
            "Password" => $password
        }
    );
}
print $mech->content();
[download]

This gets me as far as verifying that I don't have a login:

"Information entered does not match what we have on file. Please try again."

The main points here are

1. request the page you want
2. detect if you are on the login page and submit the login credentials