If you mix gases of different temperatures if you conserve Kinetic Energy for the mixed gas you end up with a gas at a different temperature than you would get if you conserved momentum for the mixed gas. For example if you mix equal amounts of Helium that are 500 degrees Kelvin with Helium that is 100 degrees Kelvin your mixed gas is 300 degrees Kelvin if you conserve KE and 261.7 degrees Kelvin if you conserve momentum. Has there ever been an experiment to find out what temperature mixing gases of different temperatures will become?

Are you asking for help with your P-Chem homework or just learning on your own? I don't think this is the best forum to help you.

First, your question as posed isn't answerable because you haven't specified the volumes and pressures of the gas mixing. Is it constant volume? Constant pressure? It isn't constant temperature, so I assume you're talking about an adiabatic process. The final temperature (and kinetic energy) depends on those things. The gases will do work and change temperature (and increase in entropy) as they expand (and compress) into each other's volumes.

In addition, I don't think your equation for total momentum is valid, as you noted. Even to get a total scalar value, you would need to add up the individual momenta of all the particles, which have velocities in a Boltzmann distribution.

To summarize, both total kinetic energy and total momentum at the macroscopic and atomic levels are conserved for all types of gas mixing and this has been very well described both experimentally and theoretically for many many years since the time of Joule, Boyle, Charles and Avogadro. This is the field of thermodynamics, which is one part of physical chemistry. It is complicated, but very important for all rocket science. If you get an answer where they aren't conserved, then you've made a mistake somewhere.

Here's a place to start. Further on, a college course in thermodynamics will go through all of this (and a lot more) in great and sometimes very painful detail. There's a reason it's sometimes referred to as "thermogoddamnics."

https://en.wikipedia.org/wiki/Adiabatic_process#Derivation_of_P-V_relation_for_adiabatic_heating_and_coolingGood luck!