To make a mint simple syrup, heat up the sugar, water and 15 mint leaves in a small saucepan over medium heat. Allow the mixture to simmer for 3 minutes and remove from the heat. Without removing the mint, chill the simple syrup for at least an hour in the refrigerator. Strain and discard the mint.
Stir the simple syrup together with the blood orange juice, lime juice, sparkling water and remaining mint leaves. Pour into the Slush & Shake Maker. Stir until slushy and then add rum. The mixture will thin out. Continue to stir until thick.