I had to have both knees rebuilt in the summer of 2012 and the doctor put a cryotherapy unit on my knees right out of surgery and they stayed on for six weeks. The unit is a cooler that holds ice water with a pump that ran to a pad that is 18" long and wraps completely around the knee.
Running one unit I would have to have a blanket on me as it would drop my body temperature quickly. It was amazing I could sit on front porch in 90 degree heat and be cold.
So, yes you can cool youself down very quickly with a unit like that. You might want to see if you can find a used cryotherapy unit and try it. The only power it uses is a 12 volt pump.