How many ways are there to divide elements into equal unlabelled sets? How many ways are there to divide N elements into K sets where each set has exactly N/K elements.
For the example of 6 elements into 3 sets each with 2 elements. I started by selecting the elements that would go in the first set (6 choose 2) and then those that would go into the second as (4 choose 2) and then the 2 remaining elements into the third set. This gives, (6 choose 2) * (4 choose 2). In general 
(N choose N/K) * (N-(N/K) choose N/K) * (N-(2*N/K) choose N/K) * ... * 1


Your approach, as noted by Ross Millikan's answer, is effective. Another way to approach such a problem would be to consider "interpreting" a permutation as such a partition - like, if we wrote the elements in the order:
$$eabcfd$$
we might just group them into pairs as $\{\{e,a\},\{b,c\},\{f,d\}\}$ - where we just "fill" the expression $\{\{\_,\_\},\{\_,\_\},\{\_,\_\}\}$ by drawing from the order in which we wrote the elements.
However, $eabcfd$ and $aebcfd$ represent the same partition, as the pairs are unordered - and $bceaf\!\,\!d$ also represents the same partition, as the order of the pairs does not matter. Proceeding thusly, we can see that we can reorder within each of the $K$ sets in $(N/K)!$ ways without affecting the partition, and we can reorder the order in which the sets appear in the partition in $K!$ ways - and that, these are the only transformations which do not affect the partition. Thus, dividing the number of permutations of the elements by the number of permutations representing any given partition yields that there are
$$\frac{N!}{(N/K)!^k K!}$$
such partitions.
(This can also be found by expanding ${a \choose b}=\frac{a!}{b!(a-b)!}$ and looking at cancellations in your expression)
