Place sugar and 2/3 cup water in a small saucepan. Bring to a boil over medium heat, stirring occasionally. Remove from heat; stir in espresso and Kahlua. Let the syrup cool.
Using a serrated knife, cut sponge cake in half horizontally, making two layers. Place one layer in the bottom of a 9-by-9-by-2-inch baking pan. Using a pastry brush, brush with 3/4 cup cooled syrup. Sift 2 tablespoons ground espresso over cake.
Place espresso ice cream in the bowl of an electric mixer. Beat on low speed with the paddle attachment until spreadable. Spread ice cream over cake; top with second layer of sponge cake. Brush with remaining syrup. Transfer cake to freezer for 20 minutes.
Remove cake from freezer; sprinkle with remaining 2 tablespoons of ground espresso. Place coffee ice cream in a mixer bowl. Beat on low speed until spreadable. Spread ice cream over cake, forming large swirls. Return to freezer; freeze until completely hardened. To serve, garnish with chocolate curls.