This does not seem a simple matrix equation to solve. Computationally, my first attempt would be with Newton's method, even if it takes $O(k^6)$ per iteration, where $k$ is the size of the matrices. The Jacobian of the map in the LHS is
$$
L_B f[H] = \sum_{i=1}^n (A_i+B)^{-1}H(A_i+B)^{-1},
$$
and to solve the equation $L_Bf[H] = Y$ for $H$ given $Y$ you need to convert it to a $k^2 \times k^2$ linear system (there are faster algorithms to solve this linear matrix equation for $n=2$, but I do not think there is anything better otherwise).

If you need to solve it for dimensions for which this is unfeasible, then I would try turning it into a fixed-point equation, for instance
$$
B = \left(K - \sum_{i=2}^n (A_i+B)^{-1}\right)^{-1} - A_1,
$$
and then hope that the iteration
$$
B_{m+1} = \left(K - \sum_{i=2}^n (A_i+B_m)^{-1}\right)^{-1} - A_1,
$$
converges.

The scalar version of this equation is a *secular equation*, but searching for this term I found nothing interesting for matrix arguments.