Add the sugar, water and mint to a small saucepan and heat over medium heat. Bring to a simmer and stir until all sugar is dissolved. Turn off the heat and allow it to sit for at least 5 minutes. Then strain it into a large pitcher. Add the lime juice and rum to the pitcher. Add the extra mint leaves and the extra lime, cut into rounds. Fill the pitcher with 4 handfuls of ice. Gently stir in soda water just before serving.