Low-Calorie Root Beer Mocha Frappe

1 (12 fluid ounce) can or bottle root beer

cold

water

or as needed

1 ¼ cups skim milk

3 tablespoons sugar-free chocolate flavoring syrup (such as Torani®)

3 tablespoons sugar-free caramel flavoring syrup (such as Torani®)

2 tablespoons instant coffee (such as Folgers®)

1 tablespoon imitation vanilla extract

1 teaspoon artificial sweetener

or to taste