Add the chopped watermelon to a blender and blend until smooth. Strain watermelon juice through a fine mesh strainer into a large pitcher. Use your fingers to press the watermelon pulp through the strainer in order to get as much of the liquid out as possible. Discard the watermelon pulp (or perhaps use it in popsicles?).

Add the remaining ingredients to the pitcher, stir, and serve over ice. Note: You can adjust the amount of added water, lemon juice, and sweetener to taste.