  We construct the limiting mixed Hodge structure of a degeneration of compact
K\"ahler manifolds over the unit disk with a possibly non-reduced simple normal
crossing singular central fiber via holonomic $\mathscr D$-modules,
generalizing some results of Steenbrink. Our limiting mixed Hodge structure
does not carry a $\mathbb Q$-structure; instead, we use sesquilinear pairings
on $\mathscr D$-modules as a replacement. The limiting mixed Hodge structure
can be computed by the cohomology of the cyclic coverings of certain
intersections of components of the central fiber. Additionally, we prove the
local invariant cycle theorem in this setting.
