In the bowl of a stand mixer, combine cake mix, pudding mix, eggs, oil, rum and egg nog. Mix on medium speed for about 3 minutes. Stir in chocolate chips until evenly dispersed in the batter.
Pour the batter into the bundt pan, evenly spreading the batter in the pan.
Bake for 40-45 minutes or until a toothpick inserted into the cake comes out clean. Remove from the oven and allow to cool for 10 minutes on a wire rack. Invert pan onto a wire rack and remove cake from the bundt pan. Cool completely and then place the cake onto a serving dish.
Prepare the glaze by placing the cream cheese in a mixing bowl and beating until light and creamy. Add powdered sugar, eggnog and vanilla and mix again until combined, adding more eggnog as needed to reach a drizzle consistency.
Fit a pastry bag with a small piping tip (or you can just cut a small corner off a plastic bag if you don’t have a pastry bag and tip), and pour in the glaze, twisting the top of your bag tightly to close. Drizzle the glaze over the cake and allow to set.
Cut into slices and serve.
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