All in all, very good invention! And also inspiring. I hope more teens will be involve in inventing things. I hope she wins. Also, (totally unrelated) I think she's cute. :)
1) a cool room at temperature T
2) enough light, L, to see by
3) a happy human at temperature H
Assuming that T,H, and L are fixed, you're going to use more energy with a less-efficient process. So, you'll 'spend' more energy to get there.
If you're trying to run the process at fixed energy, you're going to wind up accepting a slightly higher T. If that's the case, you'd be able to use even less energy by powering the LED directly and moving the setpoint for T up to that same temperature.
So to keep the temperature where you want it, you'd need to cool the room even more.
due to the temperature reliance of Peltier tiles, the flashlights worked better in colder temperatures (tested at 41 degrees Fahrenheit). The flashlight still worked in warmer temperatures (50 degrees, for instance), but the colder the surrounding temperature, the more the hand’s body heat can help the Peltier generate electricity.