This is a very interesting question. Although I am not able to come up with a full answer, nor do I know if research or established results about this topic are available, here is what I am thinking.
The incompressible inviscid flow is described by the Euler equation instead of the Navier-Stokes equation. According to the Helmholtz theorems, vortices cannot be created or destroyed in an inviscid fluid. That is, if you have thought of vortices somehow splitting into smaller and smaller ones, this does not seem to be possible, as far as I can see. This does not preclude, however, that even a single vortex line could be entangled more and more with itself (without creating knots, probably) during the evolution of the system. If it would finally become so entangled that it spreads all over the whole box, say in a fractal or Peano curve like fashion, that would be your "thermal state", I guess.
This brings us to the ergodic hypothesis. A statistical-mechanical ensemble in 6N-dimensional phase space behaves much like the incompressible inviscid flow of an ordinary fluid in 3 dimensions. The ergodic hypothesis states, that any system in the ensemble will come arbitrarily close to any other system if you wait long enough. Allegedly, and in colloquial terms, the fluid of the statistical ensemble gets naturally stirred so thoroughly that eventually it is perfectly mixed.
According to Hamilton's equations and Liouville's theorem, the ensemble represents an incompressible flow (divergence of phase space velocity is zero, hence, ensemble density is constant along the flow lines, i.e. material derivative of density is zero, and yet, the continuity equation holds). Similarly to the inviscid fluid, energy is conserved due to Hamilton. Since neighboring sample systems in the ensemble are statistically independent of each other by definition, they cannot influence each other, and so I think this is the analogy to the lack of viscosity.
Having said that, I suspect that it is difficult or impossible to prove thermalization of the ordinary incompressible, inviscid fluid, much in the same way that it has not been possible yet to prove the ergodic hypothesis of statistical mechanics (as far as I know). Moreover, there is Poincare's recurrence theorem, that any purely mechanical system under certain reasonable assumptions is almost periodical, which would preclude ergodicity.
However, I don't know if Poincare's theorem is also valid for continuum systems, particularly the incompressible inviscid fluid of your question. My intuition about stirring paint tells me that there is no almost-periodicity, but then again, this intuition has built around viscous flow, so might not be general enough.
