In a cocktail shaker, add 1 sprig of mint (about 4-5 leaves, stem removed), mint-infused simple syrup, and peach purée. Using a muddler, muddle the mint so that it breaks up in the simple syrup/peach purée juice. This should take 30 seconds to 1 minute.
Once mint is muddled, add 2 tablespoons rum and ¾ cups sprite. Shake well, pour into a glass, and garnish with two peaches and a sprig of mint.
Notes
Simple syrup and peach puree will keep nicely in the fridge so you can mix up a drink at a moment's notice.