For a smooth projective complex variety, we can generalize the classical Hodge conjecture to Beilinson’s Hodge-$D$-conjecture by considering the higher cycle map from the higher Chow groups to the Deligne cohomology. Xi Chen and James D.Lewis proved this conjecture for general $K3$ surfaces by using rational curves on singular $K3$ surfaces. On a certain type of $K3$ surfaces which can degenerate to a general hyperplane arrangement, we directly construct an enough number of higher Chow cycles to show the conjecture in an explicit way. For the proof, we consider the semi-stable degeneration given by a successive blow up of the original family to use the theory of limiting mixed Hodge structures. Since these higher Chow cycles are supported on the base locus of the family, we obtain the family of the higher Chow cycles. To show the linearly independence of the constructed higher Chow cycles, we use two different invariants of these families defined via the Clemens-Schmid exact sequence: limits and singularities of the associated higher normal functions.