Anybody know what's going wrong here?

Did you try to view the mail with another mail client? e.g pine, mutt, thunderbird? Looks like you don't have a "Content-Transfer-Encoding" mail header line (you have that in the parts only). Are you sure you want "quoted-printable" for your HTML part? you don't encode_qp() that.

Q: Could you tell me the way to Tipperary?
A: Well, I wouldn't start from here...

I would use MIME::Lite - it does all the encoding, boundary stuff and such for you.

sub sendmessage {
    my ($html,$text,$recipient,$reportdate) = @_;
    my $msg = MIME::Lite->new(
        From        => 'foo@bar.org',
        To          => $recipient,
        Subject     => "A message from me",
        Type        => 'multipart/alternative',
        Encoding    => '8bit',
    );
    $msg->attach(
        Data        => $text,
        Type        => 'text/plain',
        Encoding    => 'quoted-printable',
        Disposition => 'inline',
    );
    $msg->attach(
        Data        => $html,
        Type        => 'text/html',
        Encoding    => 'quoted-printable',
        Disposition => 'inline',
    );
    $msg->send();
}
[download]

And you don't need here-documents... ;-)

BTW, don't forget to either use or remove $reportdate.

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}