It's actually easier than you're making it:

use warnings;
use strict;
use Data::Dumper;


my %hash = (
          'ScreenName1.Description1' => {
                     'ScreenName' => 'A',
                     'Description' => 'A Description of this button',
                     'Type' => 'On/Off'
          },
          'ScreenName2.Description2' => {
                     'ScreenName' => 'B',
                     'Description' => 'A Description of this button',
                     'Type' => 'Momentary'
          },
          'ScreenName3.Description3' => {
                     'ScreenName' => 'A',
                     'Description' => 'A Description of this button',
                     'Type' => 'Momentary'
          },
          'ScreenName4.Description4' => {
                     'ScreenName' => 'A',
                     'Description' => 'A Description of this button',
                     'Type' => 'On/Off'
          },
          'ScreenName5.Description5' => {
                     'ScreenName' => 'B',
                     'Description' => 'A Description of this button',
                     'Type' => 'On/Off'
          },

);

my %ScreenNameCount;

$ScreenNameCount{$hash{$_}{ScreenName}}++ foreach keys %hash;

print Dumper \%ScreenNameCount;
[download]

This outputs:

$VAR1 = {
          'A' => 3,
          'B' => 2
        };
[download]