This should work (I think: untested code)

  my $modified_text = "";
  my $counter = 0;
  my $db_line = "";
  my $text;

  open (FH, "<", "some_file");

  while (<FH>) {
    if (m/^\s*DATABASE: /i) { # This will only match if 'DATABASE:' ap
+pears on the start of the line, or is prefixed with whitespace. /i si
+nce the case of the word probably doesn't matter.

      
      if (defined $db_line) { 
        my $insert = "WIDGETS FOUND: $counter\n"; # The text to insert
+ right after the 'database:'-line.
        $modified_data .= $db_line . $insert . $text;
      }
      $counter = 0;
      $text = "";
      $db_line = $_;
      next;
    }

    # When should the counter be increased?
    if (m/WIDGET/) { $counter++: }

    $text .= $_;
  }

  close (FH);
  
  open (FH, ">", "some_file");
  print FH $modified_text;

  if (defined $db_line) { 
    my $insert = "WIDGETS FOUND: $counter\n"; # The text to insert rig
+ht after the last 'database:'-line.
    print FH $db_line . $insert . $text;
  }
  close (FH);
[download]

Update: this approach will store all the data in the memory, if the file is too large to do that, then you should create a temp file, and change the $modified_data .= ...; line to print TEMP ...;

And then you can rename/remove the old file and rename/copy the temp file.

Update2: forgot to print the last database-section...