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Swamp coolers are a thing. I don't know about the physics, but they can actually cool a space if the ambient humidity is low.
Using ice water in the gadget you have will improve its performance, obviously.
swamp coolers work through evaporation cooling. the water absorbs some energy from the air as it evaporates. (essentially the water gets hotter, the air gets cooler.)
Too add to that, slot of that energy goes into vaporizing the water, so the average temperature is lower.
it's not technically a net loss- some of that energy is lost as it escapes the system, but conservation of energy generally means that as the air cools down, the water gets warmer. it's just that the water has extremely excellent thermal mass, meaning that the air appears to cool much more than the water gains heat. This is especially true if the water itself is cold to begin with. (ie, blowing it over ice cubes.)
This is not the dominant factor, and this system will still work even if the water is warmer than room temperature. The primary energy sink is the vaporization of water.
Vaporizing 1g of water takes 2257 Joules. https://en.wikipedia.org/wiki/Enthalpy_of_vaporization
The water does not change temperature during that process.
Heating 1 g of water 1C takes 4.184 J. To heat it from from 0C to 25C (about freezing to room temp) it thus takes104.6 J, much less than the 2257 Joules required to vaporize it. These numbers could be modified to properly account for the variace with temp, but the effect actually gets larger then I believe.
No energy leaves the system; it goes into changing the state of water and is stored in the water.