This is not right still. Both the air and water will primarily transfer heat by convection. Thermal conductivity is the measure of heat transfer within the same substance, not between two substances.

Assuming room temperature air (i belive this is the OPs intention) earlier comments are right about how the melt water will remain cold and thus have mich slower heat transfer from the ice to the water, thus slower melt.

Your missing that heat transfer is a function of the delta T (difference in temp between two susbtances) times a resistance value (convection primarily here)

Also belive this is confirmed by ice in cooler during camping. Ice and drinks stay colder if you don't drain the water.

Trick question. In Celsius you get sweaty really fast in both scenarios

Where did they go?