Preheat oven to 350.
Use an electric mixer on low speed to combine the first 6 ingredients in a large mixing bowl until it is the consistency of fine gravel, and all of the particles are almost equal in size.
Add pudding mix, eggs, pina colada mix, rum, melted butter and vanilla extract to the large bowl and combine on medium speed with an electric mixer for about 2 or 3 minutes. Scrape down the bowl halfway through. Batter should be very smooth. Fold in coconut and pineapple.
Spray a large Bundt pan with nonstick cooking spray, then pour the batter into it and smooth the top. Bake until fully golden and knife comes out clean – about 1 hr 15 minutes. Remove from oven and place on cooling rack while making rum soak.
Rum Soak: In a small saucepan combine butter, water and sugar. Carefully bring to a boil – do not allow to boil over. Reduce to a simmer and cook until sugar is dissolved and syrup becomes thicker. Remove from heat and add the rum. Mix it all up.
While cake is still cooling in the pan, pour hot syrup on top of cake. There is a lot of syrup and if cake doesn’t soak it up right away just wait a couple minutes and add the rest. It will take time to soak it up. Cool cake completely in pan before turning out onto serving platter.
For the glaze, mix icing sugar into the rum to make a thick mix. Warm it in the microwave for a few second if it gets too firm, then pour over the cake.