Bake the cake: Prepare the cake batter following the instructions on the cake mix box. Pour the batter into a 13×9-inch baking pan and bake as directed. Let the cake cool completely.
Poke and prepare the Jello: Using the handle of a wooden spoon, poke holes evenly across the top of the cooled cake. In a saucepan, bring 1 cup of water to a boil, then remove from heat and whisk in the lime Jello mix. Add 1 cup of cold water, stirring until dissolved.
Infuse the cake: Pour the Jello mixture evenly over the cake, allowing it to seep into the holes. Refrigerate the cake overnight for the best flavor infusion.
Prepare the topping: In a large bowl, whisk together Cool Whip and white chocolate instant pudding mix until fully combined. Spread evenly over the cake.
Garnish and serve: Sprinkle St. Patrick’s Day sprinkles over the top. Slice and serve chilled.