my @sorted_keys =
    map substr($_, 10),
    sort
    map sprintf('%02d%02d%5$s%02d%02d',
        m{(\d+)/(\d+) \@ (\d+):(\d+) (AM|PM)}) . $_
    keys(%line_data);
[download]

Update: Oops, I knew my AM/PM handling was wrong. Fixed:

my @sorted_keys =
    map substr($_, 10),
    sort
    map {
        my ($m,$d,$H,$M,$ampm) =
            m{(\d+)/(\d+) \@ (\d+):(\d+) (AM|PM)};
        $H %= 12;
        sprintf('%02d%02d%s%02d%02d%s', $m,$d,$ampm,$H,$M, $_)
    }
    keys(%line_data);
[download]