  This is a sequel of \cite{Wang}, which provides a general formalism for this
paper. We mainly investigate thoroughly a subclass of toric generalized
K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in
\cite{Bou}. We find torus actions on such manifolds are all \emph{strong
Hamiltonian} in the sense of \cite{Wang}. For each such a manifold, we prove
that besides the ordinary two complex structures $J_\pm$ associated to the
biHermitian description, there is a \emph{third} canonical complex structure
$J_0$ underlying the geometry, which makes the manifold toric K$\ddot{a}$hler.
We find the other generalized complex structure besides the symplectic one is
always a B-transform of a generalized complex structure induced from a
$J_0$-holomorphic Poisson structure $\beta$ characterized by an anti-symmetric
constant matrix. Stimulated by the above results, we introduce a
\emph{generalized Delzant construction} which starts from a Delzant polytope
with $d$ faces of codimension 1, the standard K$\ddot{a}$hler structure of
$\mathbb{C}^d$ and an anti-symmetric $d\times d$ matrix. This construction is
used to produce non-abelian examples of strong Hamiltonian actions.
