Preheat oven to 350°F. Spray a 12-cup bundt pan with nonstick cooking spray (the kind with flour) or butter and flour it.
Use a hand mixer or whisk to beat the cake mix, butter, eggs, water, lime juice, extract, zest, and pudding mix until a thick batter forms. Pour into prepared pan.
Bake for 35-45 minutes, or until a toothpick comes out with just a few crumbs. Remove the cake from the oven and cool for 10 minutes but keep it in the pan.
While the cake is cooling, make the sauce. Melt butter in a small saucepan over medium low heat. Stir in the sugar, zest, and juice and cook until it just simmers (bubbles around the edges). Remove from heat and stir in the vanilla.
Poke several holes in the cake using a long skewer and pour the butter sauce over the cake. Cool in pan completely, then remove to serving plate.
Make the drizzle: whisk together powdered sugar and lime juice. Drizzle over cooled cake.
Store loosely covered. Best within 2 days.