I do not have the ice cream attachment for my Kitchenaid, but I made do. The trick is to use the stand mixer whisk to stir the semi-frozen mixer every couple of hours to create a smooth, consistent final product.
Here's all you need:
1/3 cup sugar
1/3 cup light corn syrup (in the baking aisle)
1/4 cup water
2 cups seedless watermelon
1 Tablespoon lemon juice
Heat the sugar, syrup and water in a saucepan over low heat until all is dissolved. It will take less than two minutes. Remove from the heat and chill.
Remove the watermelon from the rind and place in a blend with the cleaned strawberries and lemon juice. Pulse until a smooth consistency is achieved, then press through a sieve to remove all seeds. Refrigerate.
When both components are cold, place in a frozen work bowl and use a chilled whisk to incorporate all the elements. Cover the bowl with foil and place it, along with the whisk, in the freezer. Every hour or so, whisk again to incorporate all the frozen pieces. It takes about four hours before a consistently frozen product is created.
Place in a plastic container and use within a week. It's almost like a slushie, except healthful.