Put water and sugar into a saucepan and bring them to a boil. Once boiling, add bubblegum and lower it to a simmer. Let simmer for 10 minutes. Strain the gum out of the syrup and pour syrup into a jar. Add bubblegum simple syrup and the white rum to a shaker with ice. Shake vigorously then pour into a cocktail glass. Top off with soda water and garnish with bubblegum.