In the comments above you ask how to compute these connection numbers. Paul Johnson mentions that you can do this using $S_d$ character theory. I want to also point out his two papers [Tropical Hurwitz Numbers][1] and [Chamber Structure of Double Hurwitz numbers][2], written together with Renzo Cavalieri and Hannah Markwig, are very nice and have a lot of explicit formulas. 

They are answering a question that is a bit less general than you are asking. That question is the following: Let $\sigma$ be the partition $(2,1,1,\ldots,1)$ of $n$. Let $\lambda$ and $\nu$ be other partitions of $n$. What is the coefficient of $K_{\nu}$ in $K_{\lambda} K_{\sigma}^N$? This isn't quite the same as asking about $K_{\lambda} K_{\mu}$, but you would probably learn a lot from these papers anyways.


  [1]: https://arxiv.org/abs/0804.0579
  [2]: https://arxiv.org/abs/1003.1805