When I first read your question I misinterpreted it. When I then later was about to post my answer I realized that your boldened sentence should be interpreted as "don't fiddle with anything but the match", whereas I first interpreted it as "don't fiddle with the newlines, at all". So my solution keeps the newlines and scatters out your 'jjjjjjj' to the same positions as "uekasdh" were found.

  sub replace_keep_whitespace {
    my ($str, $word, $rep) = @_;

    $str =~ join('\\n*', split //, $word);

    my $match = \substr($str, $-[0], $+[0] - $-[0]);

    my $c = 0;
    $$match =~ s/[^\n]/substr($rep, $c++, 1)/eg;

    return $str;
  }


  my $str = <<'_DATA_';
  s
  kkjn




  find
  ue
  k


  asd
  helkjadfoijejdklk
  _DATA_

  my $word = 'uekasdh';
  my $rep = 'j' x length $word;

  print replace_keep_whitespace($str, $word, $rep);

  __END__
  s
  kkjn




  find
  jj
  j


  jjj
  jekjadfoijejdklk
[download]

If the replacement word is longer than the matched word then it's simply ignored at the end. If it's shorter then, well, the remaining chars are removed (but the newlines are kept).

I realize this wasn't what was asked for, but I find the solution pretty enough to post it anyway. ;-)