Ice taken from the freezer has to be the same temperature as the inside of the freezer. When you remove it from the freezer, it will not start to melt until the interior of the cube gets up to 32 degrees.

I would advise testing any setup with an out of date bottle of insulin or an empty bottle refilled with water and a thermometer. Another way to test it would be to get one of these weather stations with a remote thermometer that keeps track of the high & low temperature and put the sensor in a plastic bag.