  We extend the Abreu-Guillemin theory of invariant K\"ahler metrics from toric
symplectic manifolds to any symplectic manifold admitting a toric action of a
symplectic torus bundle. We show that these are precisely the symplectic
manifolds admitting a Lagrangian fibration with elliptic singularities. The
base of such a toric Lagrangian fibration is a codimension 0 submanifold with
corners of an integral affine manifold, called a Delzant subspace. This concept
generalizes the Delzant polytope associated with a compact symplectic toric
manifold. Given a Delzant subspace of finite type, we provide a Delzant-type
construction of a Lagrangian fibration with the moment image being the
specified Delzant subspace. We establish a 1:1 correspondence between invariant
K\"ahler metrics and a pair consisting of an elliptic connection on the total
space of the fibration and a hybrid $b$-metric on the base Delzant subspace,
both with specified residues over the facets. Finally, we characterize extremal
invariant K\"ahler metrics as those whose scalar curvature descends to an
affine function on the base integral affine manifold. We show that this
provides a method for finding and constructing extremal K\"ahler metrics.
