Combine the water, sugar, mint, and salt in a small saucepan over medium heat. Stir until the sugar dissolves, then take the pan off the heat immediately.
Let the mint steep in the hot syrup for 10 minutes, then strain and discard the leaves. Steeping longer turns the syrup grassy and slightly bitter.
Freeze 2 cups of watermelon cubes on a sheet pan while you work. These become the ice and keep the punch from watering down.
Blend the remaining 8 cups of watermelon in batches until completely liquid. There is no need to add water.
Pour the blended melon through a fine-mesh strainer into a large pitcher or drink dispenser, pressing gently with the back of a ladle. Do not use cheesecloth — the light pulp is what gives the punch its body.
Stir in the lime juice, vodka, 3/4 cup of mint syrup, and the salt. Taste and adjust: melons vary enormously in sweetness, so add syrup a tablespoon at a time if it is thin, or lime if it is flat.
Refrigerate for at least 1 hour. The punch must be cold before the soda goes in or it will lose its fizz within minutes.
Add the club soda just before serving, pouring down the inside of the pitcher. Drop in the frozen watermelon cubes and garnish each glass with a mint sprig and a lime wheel.