You can use another structure to store the sorted results, in this case a hash of arrays of arrays:

my %sorted;

for my $key (keys %bigrams) {
    my $hash = $bigrams{$key};
    for my $secKey (sort { $hash->{$a}<=>$hash->{$b} } keys %$hash) {
        my $value = $hash->{$secKey} / $freqs{$key};
        push( @{ $sorted{$key} }, [$secKey,$value] );
    }
}

while ( my ( $key, $aoa ) = each %sorted ) {
    print "$key\n";
    for my $ary ( @$aoa ) {
        print "\t$ary->[0] => $ary->[1]\n";
    }
}
[download]

Result:

like
    VBP => 0.0882352941176471
    VB => 0.0882352941176471
    IN => 0.823529411764706
kind
    JJ => 0.2
    NN => 0.8
capped
    VBN => 0.333333333333333
    JJ => 0.666666666666667
successful
    JJ => 1
distribution
    NN => 1
[download]