If you are dealing with liquid nitrogen from air, then you are probably also dealing with liquid oxygen in the mix. I imaging that at power-storage scales the presence of such amounts of liquid oxygen might create safety issues. Separating it out might nullify any energy storage advantages.
It's not all that energetically disfavorable to extract oxygen if you're already compressing the gas. There is a lot of demand for concentrated oxygen too, so one could manufacture it as a side product instead of storing it.
As far as 'closed loop' sadly that does break the economics, I think-- part of the reason why cryogenic nitrogen storage is interesting is because the nitrogen expands 800 fold at atmospheric pressure, so the container for the decompressed nitrogen would be gigantic.
I would wager that the regulations governing LO2 generation facilities preclude pursuing that as a side product. I imagine lots of red tape, either already in place or coming soon -- right after someone blows up their neighborhood trying.
You could embed the chiller in a nitrogen only environment. I am imagining a closed loop system where you are cooling/using the nitrogen on a daily basis.
