  We give a uniform, Lie-theoretic mirror symmetry construction for the
Frobenius manifolds defined by Dubrovin-Zhang in arXiv:hep-th/9611200 on the
orbit spaces of extended affine Weyl groups, including exceptional Dynkin
types. The B-model mirror is given by a one-dimensional Landau-Ginzburg
superpotential constructed from a suitable degeneration of the family of
spectral curves of the affine relativistic Toda chain for the corresponding
affine Poisson--Lie group. As applications of our mirror theorem we give
closed-form expressions for the flat coordinates of the Saito metric and the
Frobenius prepotentials in all Dynkin types, compute the topological degree of
the Lyashko-Looijenga mapping for certain higher genus Hurwitz space strata,
and construct hydrodynamic bihamiltonian hierarchies (in both Lax-Sato and
Hamiltonian form) that are root-theoretic generalisations of the long-wave
limit of the extended Toda hierarchy.
