It's much shorter and also faster to do most of the work in XPath.

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use XML::LibXML;

my $xml = 'XML::LibXML'->load_xml(location => 'file.xml');

for my $place ($xml->findnodes('/xml/Document/Folder/Place')) {
    print $place->findvalue('Name'), "\t";
    my $name2 = $place->findvalue('ExtendedName/Data[@name="NAME_2"]')
+;
    say $name2;
}
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If you want to learn XPath, I humbly recommend you XML::XSH2, a module I happen to maintain. It's a wrapper around XML::LibXML which features an interactive shell where you can play with the data and XPath. The above code can be tried like this:

$scratch/> open file.xml
parsing file.xml
done.

/> for /xml/Document/Folder/Place echo (Name) ExtendedName/Data[@name=
+"NAME_2"]
Location 1 Salt Lake City
Location 2 Providence
Location 3 Green Bay
Location 4 Casper
[download]