  The equivariant $K$-theory of the semi-infinite flag manifold, as developed
recently by Kato, Naito, and Sagaki, carries commuting actions of the
nil-double affine Hecke algebra (nil-DAHA) and a $q$-Heisenberg algebra. The
action of the latter generates a free submodule of rank $|W|$, where $W$ is the
(finite) Weyl group. We show that this submodule is stable under the nil-DAHA,
which enables one to express the nil-DAHA action in terms of $W\times W$
matrices over the $q$-Heisenberg algebra. Our main result gives an explicit
algebraic construction of these matrices as a limit from the (non-nil) DAHA in
simply-laced type. This construction reveals that multiplication by equivariant
scalars, when expressed in terms of the Heisenberg algebra, is given by the
nonsymmetric $q$-Toda system introduced by Cherednik and the author.
