As far as I know, Gallavotti proved the ergodicity of the Lorentz gas, while Sinai proved that of a system of $N \leq 5$ rigid spheres. Anyway, this is a minor detail. For certain aspects, a more suitable model for the Drude model is the Boltzmann gas. Lanford has shown (in 1970s, I think) that the entropy for this model is always increasing, but anyone has proved that Boltzmann gas is ergodic. So the answer to your question is: if, at our level of accuracy, Lorentz gas was an appropriate mathematical scheme for the Drude model, than it would be ergodic. Else we can't conclude, since Sinai's result is very important but too limited. (at the present day.)
However, it is an interesting question from a mathematical point of view (for me, for example, it really is), but for a physicist it is not very important, since Drude model does not provide an appropriate level of precision for most of calculations carried out nowadays in solid state physics (at least, concerning what I have seen in my course of condensed matter, I'm not a specialist in solid state physics). Moreover, any of that models takes in account coulombian interactions between charged particles. (this remark restricts - in principle - a lot the domain of applicability of such schematizations.)
I think you could find very interesting the treatments of this subject (ergodic theory) by Halmos and Arnold in their classical monographs.
P. R. Halmos, Lectures on ergodic theory
V.I. Arnold, Ergodic problems of classical mechanics and Mathematical methods of classical mechanics
G. Gallavotti, Statistical mechanics and The elements of mechanics