Just as importantly, how do we measure the heat that's transported from the surface layer into the twilight zone and deep oceans?
I think the simplest answer is through water mass formation, the resulting thermohaline circulation, mixing caused by wind predominantly in shallow regions, and through downwelling processes. The heat exchanged with the atmosphere in the surface layers determine the temperature of the near-surface water. From there, mixing (caused by wind, for instance) will transfer some of the heat to lower layers in the ocean. In the case of areas of water mass formation, the sinking of surface waters is usually enhanced. When the water masses are away from the ocean boundaries (surface and bottom), then the thermohaline circulation controls the transport and exchange between water parcels and thus the heat transfer.
Solar uv is an extremely effective means of heating a body of water, the atmospheric temperature in comparison is ineffective.
Water has a high heat capacity, as uv Penetrates deeper into the water it's acts on a greater volume and transfers a lot of energy.
The ARGO drifting bouy programme has data accessible to the public which gives us data down to 2000m pretty much worldwide. This is a great resource! But in terms of deep ocean I think we are pretty much in the dark.
Thermohaline circulation is indeed a major route for surface water to overturn to the depths. Ssta and argo will give a good idea of what is heading down there.