You can use grep to pick a member from an array. You can use map to turn the list of hash refs to a list of values stored inside of them. 
#!/usr/bin/perl

use strict;
use LWP;
use XML::Simple;
use Data::Dumper;

my $var_lat = "45.96968";
my $var_lon = "-93.33434899999997";

my $ua = LWP::UserAgent->new;
my $forecast_xml = $ua->request(HTTP::Request->new(GET => "http://fore
+cast.weather.gov/MapClick.php?lat=$var_lat&lon=$var_lon&unit=0&lg=eng
+lish&FcstType=dwml"));
my $forecast_arr = XMLin($forecast_xml->content);

my ($item) = grep 'k-p12h-n13-1' eq $_->{'layout-key'},
                  @{ $forecast_arr->{data}[0]{'time-layout'} };
my @names = map $_->{'period-name'}, @{ $item->{'start-valid-time'} };
print "@names\n";

[download]

Update: The structure you are dealing with is not a hash of hashes, but an array of hashes (that's why you were able to iterate for over it). VAR1 is not a key, it is just what Data::Dumper adds to the output.

لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ

In reply to Re: Nested hashes, non-unique names by choroba
in thread Nested hashes, non-unique names by kthelen

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