Lines are ending with '\r\n', so the following code might help:
open FILE, '<:crlf', 'index.html'

Anyway, I would suggest the following script:

#!/usr/bin/perl

use strict;
use warnings;

require LWP::UserAgent;
my $ua = LWP::UserAgent->new(
                             keep_alive    => 1,
                             env_proxy     => 1,
                             show_progress => 1,
                            );

my $log_file = 'ipchanges.log';
open my $data_fh, '>>', $log_file or die "Unable to append to $log_fil
+e: $!";

my $old_fh = select($data_fh);
++$|;
select $old_fh;

my $old_ip = '';    # your current ip

while (1) {
    my $ip = '';
    if ($ua->get('http://whatsmyip.net')->content =~ m{^<title>What's 
+My IP - Your IP is: ([\d.]+)}mi) {
        $ip = $1;
        printf {$data_fh} "***%s***@***%s***%s***\n", scalar localtime
+(time), $ip, $ip eq $old_ip
          ? 'SAME'
          : 'CHANGED';
    }
    $old_ip = $ip;
    sleep 60;
}
[download]