I can't seem to find a public daytime server

A Google search for public daytime server rapidly finds the National Institute of Standards and Technology, which runs NTP and DAYTIME servers.

See Set Your Computer Clock Via the Internet and NIST Internet Time Servers.

Your next challenge is to write a DAYTIME server, and configure launchd on your own system to run it on demand.

The following modification of your code will parse the NIST standard DAYTIME format.

use strict;
use warnings;
#use 5.010;

use IO::Socket;

#my $host = 'localhost';
#my $host = 'time.nist.gov';
#my $host = 'nist1-sj.ustiming.org';
my $host = 'time-nw.nist.gov';      # pester microsoft
my $port = 'daytime(13)';

my $sock = IO::Socket::INET->new(
    Proto       => 'tcp',
    PeerHost    => $host,
    PeerPort    => $port,
) or die "cannot connect: $!";

while (<$sock>) {
    s/^[[:space:]]+//;
    s/[[:space:]]+\z//s;
    next if /^$/;

    # NIST Daytime Protocol format
    # http://tf.nist.gov/service/its.htm
    # JJJJJ YR-MO-DA HH:MM:SS TT L H msADV UTC(NIST) OTM
    if ($_ =~ /^(\d+)\s+(\d{2})-(\d{2})-(\d{2})\s+(\d{2}):(\d{2}):(\d{
+2})\s+(\d{2})\s+(\d)\s+(\d)\s+([0-9\.]+)\s+UTC\(NIST\)\s+(.)/)
    {
        my ($mjd, $yr, $mo, $da) = ($1,$2,$3,$4);
        my ($hour, $min, $sec, $dst, $lsec) = ($5,$6,$7,$8,$9);
        my ($health, $msADV, $otm) = ($10,$11,$12);
        print "NIST: $yr-$mo-$da $hour:$min:$sec\n";
    }
    else
    {
        print "$_\n";
    }
}
[download]