In triangle $ABC$, $\angle ABC = 54^\circ$ and $\angle ACB = 42^\circ$. Point $D$ is the foot of the altitude from vertex $A$ to $BC$, and $I$ is the incenter of $\triangle ABC$. Point $K$ lies on line $AD$, such that $D$ is between $A$ and $K$ and $AK$ is equal to the diameter of the circumcircle of $\triangle ABC$. Find the measure of $\angle KID$.