I don't see what's wrong with yours. I found it easier to write my own.

use strict;
use warnings;

use Data::Dumper;

my %global;
$global{'title'}='Report for today';
$global{'data 1'}{title}='My data 1 title';
$global{'data 1'}{data}= {};
$global{'data 1'}{footer}='My footer';
$global{'data 2'}{title}='My data 2 title';
$global{'data 2'}{subtitle}='Data 2 subtitle';
$global{'data 2'}{data}= {};
$global{'data 2'}{'data percentages'}= {};
$global{'data 2'}{'data raw'}= {};
$global{'data 2'}{footer}='Footer for data 2';

unspace_keys( \%global );
print Dumper \%global;

sub unspace_keys {
    my $hash_ref = shift;

    foreach my $spaced_key ( grep /\s/, keys %{ $hash_ref } ) {
        ( my $new_key = $spaced_key ) =~ s/\s+//g;
        if ( exists $hash_ref->{$new_key} ) {
            die "unspaced key '$new_key' exists";
        }
        $hash_ref->{$new_key} = $hash_ref->{$spaced_key};
        delete $hash_ref->{$spaced_key};
    }

    unspace_keys( $_ ) for grep { ref $_ eq ref {} }
                                values %{ $hash_ref };

    return;
}
__END__
$VAR1 = {
          'data2' => {
                       'subtitle' => 'Data 2 subtitle',
                       'data' => {},
                       'title' => 'My data 2 title',
                       'dataraw' => {},
                       'datapercentages' => {},
                       'footer' => 'Footer for data 2'
                     },
          'title' => 'Report for today',
          'data1' => {
                       'data' => {},
                       'title' => 'My data 1 title',
                       'footer' => 'My footer'
                     }
        };
[download]

This is (also) a recursive solution. It does not check for cycles, so it will need some work if your input has any. It does check if a new unspaced key already exists, so it won't clobber anything if you have keys like 'psychotherapist' and 'psycho therapist'.