For a finite universal set $U$, let $P(U)$ denote its power set. For any two fixed subsets $A, B \subseteq U$, we define a family of subsets: $\mathcal{K} = \left\{ X \in P(U) : X \cap A \subseteq B \right\}$ If $|U| = 10$, $|A| = 6$, $|B| = 5$, and $|A \cup B| = 8$, determine the cardinality of $\mathcal{K}$.