Prepare cake mix according to package directions for two 8-inch round layers. Let cool completely. Frost cake with vanilla frosting, and place on serving plate. Score top of cake into eight wedges. Cover all but one wedge with wax paper. Cover exposed wedge with solid layer of sprinkles (in one color), then cover with wax paper. Remove wax paper from next wedge, and cover with sprinkles (in next color). Repeat for all colors. Using all eight colors, pour sprinkles around bottom edge of cake.
Knead flour into cookie dough. Prepare according to package directions, cutting out two dozen 1-1/2-inch cookies. Bake and let cool. Thin royal icing with 1 teaspoon water. Using fingertip, spread icing over 12 cookies. Cover each cookie with solid layer of sprinkles or sugar. Let set 30 minutes. Press cookies into side of cake. Let kids decorate remaining cookies.
In bowl, with electric mixer on low, beat together all ingredients until smooth. Increase mixer speed to high, and continue beating until stiff peaks form.