Answer
Let L represent the heat of fusion of ice, 335,000 J/kg.
Using the hint in the problem, we set the initial potential energy equal to the energy needed to melt that mass of ice. Notice that the mass cancels out.
$$mgh = mL$$
$$gh = L$$
$$h = \frac{L}{g} = \frac{335000 J/kg}{9.8 \frac{m}{s^{2}}} = 34184 m = 34 km$$
