  We construct the limiting mixed Hodge structure of a degeneration of compact
K\"ahler manifolds over the unit disk with a possibly non-reduced normal
crossing singular central fiber via holonomic D-modules, which generalizes some
results of Steenbrink. Our limiting mixed Hodge structure does not carry a
Q-structure; instead we use sesquilinear pairings on D-modules as a
replacement. The associated graded quotient of the weight filtration of the
limiting mixed Hodge structure can be computed by the cohomology of the cyclic
coverings of certain intersections of components of the central fiber.
