In a medium saucepan, combine the sliced ginger with the smashed lemongrass and 4 cups of the water and bring to a boil over high heat. Stir in the sugar. Cover and let the ginger-lemongrass syrup steep off the heat until cool, about 2 hours.
Strain the syrup through a fine-mesh sieve. Discard the ginger and lemongrass. Stir in the lemon juice and the remaining 2 cups of water. Refrigerate until chilled.
Make the soda according to the manufacturers' directions. Serve in tall glasses over ice, garnished with lemon wheels and lemongrass stalks.