Put the tea bags in a large heatproof pitcher and pour in 5 cups boiling water; let steep 10 minutes, then remove the tea bags. Pour about half of the tea into ice cube trays and freeze until firm, about 3 hours. Refrigerate the remaining tea.
Combine the sugar and 1 cup water in a saucepan; simmer over medium-low heat, stirring, until the sugar dissolves. Remove from the heat and let cool completely.
Stir 1 cup of the sugar syrup, the lemon juice, 5 cups cold water and the vodka into the chilled tea, then add the tea ice cubes. Pour into glasses; serve with lemon slices and mint leaves.