The approach noted in the Question of computing $(nk)!/n!$, which is exact if all the items are distinct, in general gives an upper bound on the counting. A modest change in computation gives us a lower bound:
Let the multiplicities of distinct items be $m_1,m_2,\ldots,m_t$, so that: $$ \sum_{i=1}^t m_i = nk $$ Then the desired count will be at least: $$\frac{(nk)!}{m_1!m_2!\ldots m_t! n!} $$
One way to realize this is a lower bound is by choosing an arrangement of all $nk$ items. By the multinomial formula there are $ \binom{nk}{m_1,m_2,\ldots,m_t}=\frac{(nk)!}{m_1!m_2!\ldots m_t!} $ ways, which might be thought of as the "single block" case $n=1$. Now snip the single block into a sequence of $n$ equal sub-blocks. If all the blocks were distinct, then dividing by $n!$ gives the exact corresponding number of arrangements, but since in general there could be identical sub-blocks, this can be an over-reduction. Thus the above is only a lower bound.
In the example provided in the Question this does give us an exact count:
$$ \frac{4!}{1!2!1!2!} = 6 $$
because it is impossible with those multiplicities to get two identical sub-blocks.
This approach can be modified (with additional effort required) to give an exact count. We introduce some terminology to aid in formulation.
Definition Let $\mathscr{S}$ be a multi-set of $k$-permutations drawn from multi-set $A$ whose separate multiplicities add up to the corresponding total multiplicities in $A$. We will say that $\mathscr{S}$ is a $k$-tuple resolution of $A$, but this is just to coin a phrase.
For each $k$-resolution $\mathscr{S}$ of $A$, we associate with it $\sigma_\mathscr{S}$ an integer partition of $n$ given by counting the multiplicities of the $k$-permutations belonging to $\mathscr{S}$. If all the $k$-permutations in $\mathscr{S}$ are distinct, then $\sigma_\mathscr{S}$ is simply the sum-of-ones $1+1+\ldots+1=n$. Otherwise the partition $\sigma_\mathscr{S}$ will have some part greater than $1$.
Let $\Sigma = \sigma_\mathscr{S}$ be a random integer partition of $n$ defined by randomly sampling $\mathscr{S}$ as outlined in the above, splitting a random permutation of $A$ into a sequence of $k$-permutations, which are then collected into the multi-set $\mathscr{S}$.
Let $\mu(\sigma) = \frac{n!}{p_1!p_2!\ldots p_s!} = \binom{n}{p_1,p_2,\ldots,p_s}$ be the multinomial coefficient corresponding to integer partition $\sigma$ of $n$ with $s$ parts:
$$ p_1 + p_2 + \ldots + p_s = n $$
If $\mathbf{P_n}$ is the set of all integer partitions of $n$, the probability distribution of $\Sigma$ determined by the above sampling produces probability values $p_\sigma = \Pr(\Sigma=\sigma)$ for each $\sigma \in \mathbf{P_n}$.
The exact count of distinct $k$-resolutions of $A$ is then:
$$ \frac{(nk)!}{m_1!m_2!\ldots m_t!} \sum_{\sigma \in \mathbf{P_n} } \frac{p_\sigma}{\mu(\sigma)} $$
The difficulty in computing the probabilities $p_\sigma$ depends on the multiplicities $m_i$ of items in $A$. When $k$ is large and a high fraction of the items belong to the classes of items with small multiplicities, then the probability $p_\sigma$ will be concentrated on the sum-of-ones integer partition $\sigma$. Then our lower bound above will be correspondingly close to the actual value.
References
I'm unable as yet to give a citation that considers the OP's exact problem. It concerns "partitioning" a multiset, not into sets or multisets, but into equal length $k$-permutations (aka $k$-tuples). The latter topic has been discussed here and is fairly standard in combinatorial terminology.
Lacking a more specific reference I will point out a pair of book length (or longer) treatments. One of these is also mentioned in previous posts:
Stanley, Richard P. Enumerative Combinatorics, (Cambridge University Press, 1997/1999): two volumes
MacMahon, Percy A. Combinatory Analysis, (AMS Chelsea Publishing, 2001): two volumes bound as one book
The latter is a reprint of volumes by MacMahon originally published in 1915 and 1916. This material was also covered in a 1978 work edited by George Andrews:
Andrews, George E. (editor) Percy Alexander MacMahon: Collected papers, Volume I, Combinatorics (Mathematicians of Our Time Series, The MIT Press, Cambridge, Massachusetts, 1978)
Andrews' book seems now out of print, but I suspect it is a very desirable way to appreciate MacMahon's work, esp. on partitions of multisets. A 1980 review by John Riordan in Bull. AMS is available online by Project Euclid. The Internet Archive also has images of MacMahon's original volumes.
