It is slow probably because you seem to be doing an awful lot of reading and writing. An example log entry would have helped, but the following (with the addition of the appropriate checking) will create the log file on sorted order without a temp file and without readin the whole file into memory.

my $MAXBUFLEN = 1024 * 1024; # 1M

sub sort_logfile {
  my $file = shift;
  my @off = (0);
  my @lines;
  open(INFILE, "<$file");
  while(<INFILE>) {
    push @off, tell;
    if (/^\d.*?\[([^\]]+)\]/) {
      push @lines, [$1, $.-1];
    }
    else {
      # assume any non-matching line
      # is a continuation of the previous
      $off[-2] = $off[-1];
      next;
    }
  }

  @lines = sort { $a->[0] cmp $b->[0] } @lines;

  open(INFILE,"<$file");

  my $line = 0;
  my $buf;
  while (my $i = shift @lines) {
    my ($start, $end) = ($i->[1],$i->[1]+1);
    while( @lines and $lines[0]->[1] == $end) {
      shift @lines;
      $end++;
    }
    ($start,$end) = @off[$start,$end];
    my $len = $end - $start;
    while ($len > 0) {
      seek(INFILE, $start, 0) or die "seek: $!";
      read(INFILE, $buf='', $len > $MAXBUFLEN ? $MAXBUFLEN : $len)
        or die "read: $!";
      print $buf;
      $len -= length $buf;
    }
  }
}
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