Preheat oven to 350 degrees F (175 degrees C). Grease a 10-inch tube pan.
Beat egg whites in a glass or metal bowl until soft peaks form. Lift your beater or whisk straight up: the egg whites will form soft mounds rather than a sharp peak.
Beat 2 cups confectioners' sugar and 1 cup butter together in a bowl until creamy; add egg yolks, 1 at a time, until fully incorporated.
Mix flour and baking powder together in a bowl. Stir flour mixture into creamed butter mixture, alternating with cooled coffee until just mixed; mix in sprinkles. Fold egg whites into batter. Pour batter into the prepared pan.
Bake in the preheated oven until edges of cake pull away from sides and a toothpick inserted in the center comes out clean, about 1 hour. Cool cake in pan for 1/2 hour before turning onto a serving plate.
Beat 2 cups confectioners' sugar, 1/2 cup butter, and cocoa powder together in a bowl until smooth; beat in hot coffee and vanilla extract until frosting is creamy. Spread frosting onto cooled cake.