Cream shortening; gradually add sugar, beating well.
Combine flour, baking powder, and salt; add to creamed mixture alternately with milk, beginning and ending with flour mixture. Stir in coconut and vanilla.
Beat egg whites (at room temperature) until soft peaks form; fold into batter.
Spoon batter into 2 greased and floured 9-inch round cake pans. Bake at 350° for 35 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes; remove layers from pans, and let cool completely. Spread Orange Coconut Filling between layers; spread top and sides of cake with White Frosting.