  Let $k$ be a number field. We give an explicit bound, depending only on
$[k:\mathbf{Q}]$ and the discriminant of the N\'{e}ron--Severi lattice, on the
size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic
to the Kummer surface attached to a product of isogenous CM elliptic curves. As
an application, we show that the Brauer--Manin set for such a variety is
effectively computable. Conditional on GRH, we can also make the explicit bound
depend only on $[k:\mathbf{Q}]$ and remove the condition that the elliptic
curves be isogenous. In addition, we show how to obtain a bound, depending only
on $[k:\mathbf{Q}]$, on the number of $\mathbf{C}$-isomorphism classes of
singular K3 surfaces defined over $k$, thus proving an effective version of the
strong Shafarevich conjecture for singular K3 surfaces.
