Place a rack in the center of the oven and preheat the oven to 350 degrees. Lightly grease three 9-inch round cake pans with solid vegetable shortening, then dust with flour. Shake out the excess flour. Set the pans aside.
Place the cake mix, strawberry gelatin, mashed strawberries and juice, oil, milk and eggs in a large mixing bowl and blend with an electric mixer on low speed for 1 minute. Stop the machine and scrape down the sides of the bowl with a rubber spatula. Increase the mixer speed to medium and beat for 2 minutes more, scraping the sides down again if needed. The strawberries should be well blended into the batter. Fold in the coconut and pecans. Divide the batter among the prepared pans and place them in the oven; if your oven is not large enough, place two pans on the center rack and place the third pan in the center of the highest rack.
Bake the cakes until they are light brown and just start to pull away from the sides of the pan, 28 to 30 minutes. Be careful not to overcook the layer on the highest oven rack. Remove the pans from the oven and place them on wire racks to cool for 10 minutes. Run a dinner knife around the edge of each layer and invert each onto a rack, then invert again onto another rack so that the cakes are right side up. Allow them to cool completely, 30 minutes more.
Meanwhile, prepare the frosting. Combine the cream cheese and butter in a medium bowl with an electric mixer on low speed for about 30 seconds. Stop the machine and add the sugar and drained strawberries. Blend the frosting on low until the sugar has been incorporated. Then raise the speed to medium and mix the frosting another minute or until the frosting lightens and is well combined. Fold in the coconut and pecans.
To assemble, place one cake layer, right side up, on a serving platter. Spread the top with frosting. Add another cake layer, right side up, and frost the top. Repeat this process with the third layer and frost the top. Use the remaining frosting to frost the sides, working with clean, smooth strokes. Serve at once or chill the cake for later serving.
Place this cake uncovered, in the refrigerator until the frosting sets, 20 minutes. Cover the cake with waxed paper and store, in the refrigerator, for up to 1 week. Or freeze it, wrapped in aluminum foil, for up to 6 months. Thaw the cake overnight in the refrigerator before serving.