Most delicious and awesome drink, You would want it a second and third helping.
It shouldn't become too squashed like grinding it in a mixie. But if you preferred to do so you can do it if you don't want to use your hand!
There is no exact measurement of water, If the melons are very sweet you don't need a lot of Jaggery.
Jaggery comes in round blocks or square blocks, grate or mash them up and use them as you require depending on the sweetness of the melons. Jaggery/Gaur is a sweetener used in India, it is not a processed sugar and is the raw product of cane and is a better option than sugar.