$$K_{ik}=-\frac{\partial}{\partial x_i}\int\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)dy$$
$$\quad=\int\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)dy$$

now use that

$$\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}=-\frac{4\pi}{3}\delta(y-x)\delta_{ik}-\delta_{ik}|y-x|^{-3}+3(y_i-x_i)(y_k-x_k)|y-x|^{-5}$$

the Dirac delta function appears because of the Laplacian identity $\Delta |y|^{-1}=-4\pi\delta(y)$ (which incidentally holds only in three dimensions)

some more steps, at the request of the OP; note the identity
$$\int_{|y-x|<d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}dy=\frac{1}{3}\delta_{ik}\int_{|y|<d}\Delta |y|^{-1}\,dy=-\frac{4\pi}{3}\delta_{ik}$$
then decompose the integrals in the definition of $K$ into
$$K_{ik}=\int_{|y-x|<d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)\,dy+\int_{|y-x|>d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)\,dy$$
$$\quad=\mu(x)\int_{|y-x|<d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\,dy+\int_{|y-x|<d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}[\mu(y)-\mu(x)]\,dy+\int_{|y-x|>d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)dy$$
$$\quad=-\frac{4\pi}{3}\delta_{ik}\mu(x)+\int_{|y-x|<d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}[\mu(y)-\mu(x)]\,dy+\int_{|y-x|>d}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|}\mu(y)dy$$

**higher dimensional generalizations:**

the OP asks in a comment for the more general integral

$$K_{ik}^{(n)}=\int_{\mathbb{R}^n}\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_k}\frac{1}{|y-x|^{n-2}}\mu(y)d y$$

where now $x,y\in\mathbb{R}^n$ and $n\geq 3$. This can be evaluated in the same way, using the $n$-dimensional generalization of the Laplacian identity,

$$\Delta_n|r|^{2-n}=-(n-2)S_n\delta_n(r),\;\;n\geq 3$$

with $S_{n}$ the surface area of the $n$-dimensional unit sphere.