First challenge: It is suggested in the documentation for Mail::Internet that you use MIME::Entity for multipart support. Mail::Internet and Mail::Field are explicitly not going to help you in this context. You already use MIME::Head, so I'm going to assume you have MIME::Tools installed.

Second challenge: Test data would be really nice. But hey, I had to look up how to do it myself, so no points against you. I'm not even sure I'm doing it right myself. I generated a test email and attached it to the code in the __DATA__ chunk.

use warnings;
use strict;
use MIME::Parser;

my $parser = MIME::Parser->new();

# The message and attachment are extracted to disk, so there will be a
# couple of files in this directory.
$parser->output_under('/tmp');
my $entity = $parser->parse(\*DATA);

if ( $entity->is_multipart ) {
    # This message is multipart, so look at each part
    for my $part ( $entity->parts ) {
        my $head = $part->head;
        # We only care about attached content in this example.
        if ( my $filename = $head->recommended_filename ) {
            print "Attachment name: $filename\n";
        }
    }
}

__DATA__
Content-Type: multipart/mixed; boundary="----------=_1228756284-7253-0
+"
Content-Transfer-Encoding: binary
MIME-Version: 1.0
X-Mailer: MIME-tools 5.427 (Entity 5.427)
From: me@myhost.com
To: you@yourhost.com
Subject: Hello, Nurse!

This is a multi-part message in MIME format...

------------=_1228756284-7253-0
Content-Type: text/plain; name="short.txt"
Content-Disposition: inline; filename="short.txt"
Content-Transfer-Encoding: binary

This is a short text file

------------=_1228756284-7253-0
Content-Type: text/plain
Content-Disposition: inline
Content-Transfer-Encoding: binary

some literal text
------------=_1228756284-7253-0--
[download]

Hopefully you can fill in the remaining blanks for your app, or somebody else can show an even more streamlined way to do it. Have fun!