I think that Convert::PEM can do what you're looking for. See the code snippet below borrowed from stuff-things.net Use the return value from decryptPEM($file,$password); as the value of your $key_string variable in your code.

#
# borrowed from
# http://stuff-things.net
#
use Convert::PEM;
sub decryptPEM {
  my ($file,$password) = @_;

  my $pem = Convert::PEM->new(
      Name => 'RSA PRIVATE KEY',
      ASN  => qq(
          RSAPrivateKey SEQUENCE {
          version INTEGER,
          n INTEGER,
          e INTEGER,
          d INTEGER,
          p INTEGER,
          q INTEGER,
          dp INTEGER,
          dq INTEGER,
          iqmp INTEGER
              }
    ));
    my $pkey =
      $pem->read(Filename => $file, Password => $password);

    return(undef) unless ($pkey); # Decrypt failed.
    $pem->encode(Content => $pkey);
}
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