"The time a gas takes to reach equilibrium" isn't really well-defined -- and even trying to interpret it in terms of an average sense can be tricky: in particular, there may very well be states which never reach equilibrium (consider very carefully placed and aimed gas moleculres in a box). As far as physics is concerned though, the <PERSON> equation is what's believed to govern the approach to equilibrium, but one must put in some assumed "collision term" to actually put it to use.
In terms of where your question may have been studied in current research, the issue of thermalization is obviously going to be a bigger issue in fields of physics when one is dealing with numbers of particles near the limits of validity of thermodynamics, so that's a sign that you might look for papers in the field of "cold atoms". See for example this 2001 paper by the group of <PERSON> which carefully measures a "thermalization time" $\tau_{th}$ which they define via a <PERSON> equation-inspired model inspired by a theory paper of <PERSON> et al. 682
These papers might be a good place to start to see how working physicists think about thermalization.
You frame your question in terms of "ideal gases" but as you'll notice from these papers, when you get down to numbers of particles and temperatures which are manageable directly in these experiments, quantum effects play a nontrivial role in the population dynamics. (It's also worth giving some thought to the following: how can an ideal gas equilibrate if the particles are defined not to interact with each other? Your answer will probably help you figure out better what exactly you're looking for.)
If you're more interested in what happens to a large number of "classical" particles and how they thermalize purely through collisions, this is fairly easy to simulate on the computer with molecular dynamics code, and then you can measure whatever you like with the computed particle trajectories. Of course this isn't strictly a physical "experiment" but it is in my opinion a real test of the kinetic theory predictions all the same. 28