You can do this task all sorts of ways in perl. It is often done using what is know as "inplace editing". It is in the FAQ which are worth a look if you are new to perl. Inplace Editing shows several idioms and one liners. The classic is:

perl -pi.bak -e 's/OLDSTRING/NEWSTRING/g' FILELIST
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Taking the code you have, adding some error checking and rationalising your matching regex where you capture things into $1 and $2 but do nothing with them (untested):

for my $file (@files){
  open IN,"$directory/$file" 
    or die "Can't read $directory/$file Error: $!\n";
  open OUT,">$directory/$file.out" 
    or die "Can't write $directory/$file.out Error: $!\n";
  while(<IN>){
    chomp;  # remove newline so new data on same (?correct?) line
    print OUT $_;
    print OUT "\t\t <local-id>$file</local-id>" if /<asset-id/;
    print OUT "\n";
  }
  close IN;
  close OUT;
  # now unlink IN and rename OUT to IN if desired.
  # warning unlink will delete the original file
  # unlink "$directory/$file" or die $!;
  # rename "$directory/$file.out", "$directory/$file" or die $!;
}
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This is not an elegant of perlish way of doing it but should get you heading in the right direction.