This may not be optimal for your situation, but it is possible to change your linebreak value to something more suitable to your needs;

#Untested
{ # localize the block
  local $/ = q{ } # space character is the new newline
  my $fh;
  unless (open $fh, "<".$logfile_name) {
    die ("File open failure\n\t$!\n");
  }
  my $buffer;
  while (<$fh>) {
    next if /\n/;
    if (/^\d\d:\d\d:\d\d/) {
      print $buffer." ";
      $buffer = $_;
    }
    else {
      $buffer .= $_." ";
  }
  print $buffer;
  close $fh;
}
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