What is needed to calculate the amount of heat absorbed as a substance melts?
It would usually be done the other way around with a mass of known thermochemical properties and an accurate thermometer in a well-insulated container.
Measuring heat input to a particular mass is difficult due to losses to the system and surroundings.
More practically a weighed sample of material would be melted, and its temperature held just at its melting point. Then it would be immersed in a liquid (water or oils, depending on the properties and temperatures involved) with a known mass and temperature. The heat energy released to freeze the material can then be calculated from the temperature rise of liquid.
On the expensive side, a Differential Scanning Thermography (DST) instrument could be used to accurately measure the heat input directly to the sample to the melting point.