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.