Just remember whether you've encountered "***Private-from-iPhone***" (say in some variable $is_private), and only add the line in this case:

...
my $is_private = 0;
while (<>) {
  $is_private = 0 if eof;  # reset 
  chomp;
  if (/\Q***Private-from-iPhone***/) {
    $is_private = 1;
  }
  # print additional line
  if ($is_private && /SUMMARY:/) {
    s/$/\nX-CALENDARSERVER-ACCESS:CONFIDENTIAL^M/;
  }
  print "$_\n";
}
...
[download]

(This assumes that DESCRIPTION: occurs before SUMMARY:, of course)

Update: added # reset line, as per GrandFather's/moritz's note — (I had overlooked that the script is supposed to process multiple files in one go).