This is a good start, but it's going down the path of calculating heat flux, or how fast heat can transfer between two bodies (via convection or whatever mode.) I think it's like trying to solve billiard ball problems using F=Ma, when the easier approach is conservation of energy (1/2 MV^2).

Now that I'm sober with a bit of coffee, maybe we could set up the problem like this to make it easier:

Assume we take two identical glasses chilled at 0 deg C, add one lump of ice with different shapes of ice (sphere and cube) to each, also frozen to exactly 0C. Now add whiskey that is also from the same freezer at 0C. This is all mixed in the walk-in freezer, where the air is also 0C.

If the above two drinks are left sitting there, they would each remain in stasis, with no ice melting, right? Right.

Now we pick each glass up and hold it in our bare hands for a while, swirling it allowing 1 kJ of energy to transfer to the glass/whiskey/ice system. Assuming all of that energy makes it to the ice, how much melts? (If you want to nit-pick, let's say 2 kJ goes into the glass, then 1kJ radiates/convects back out of the glass to the atmosphere, the other 1 is cooled by the ice.)

Here's the formula I was looking for.

q = m·ΔHf

1000 J = m x 334 J/g m =~ 3g

Note the amount of ice melted is independent of the shape.

If we take the drinks out of the freezer and out on the veranda, the heat flux into the glasses could be a difficult calculation, but it is the same for either shaped cube, as they are both submerged within the fluid.