As others pointed out in the comments, the topological Hochschild cohomology of $K(n)$ (i.e., the center of $K(n)$) was calculated by Angeltveit. Here is the paper: https://doi.org/10.2140/gt.2008.12.987 Angeltveit's answer is that the center of $K(n)$ is the Morava $E$-theory spectrum $E_n$ or some extension thereof, depending on the $A_{\infty}$-structure on $K(n)$ that you began with.