You almost have it. Simply add these small changes:

print_hash(\%hash,'');
#                ^^^
sub print_hash {

    my($href,$pre) = @_;
#           ^^^^^
    foreach my $key (keys %$href) {
    if (ref($$href{$key}) eq 'HASH') {
        my $href2 = $$href{$key};
        print_hash($href2,$pre.$key.'/');
#                        ^^^^^^^^^^^^^^
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It's just a matter of letting the "recursed" calls now of the part which ist to $prepend ;-)