The cake is a dense but fluffy with lots of texture from the shredded coconut. Although it seems fiddly to do a cake, syrup and frosting, it is actually easy to make. You can eliminate the frosting but do cover the cake in the lime syrup because it creates a more intense lime flavor.
Cake:
225 grams butter, at room temperature
2.5 dl sugar
1 teaspoon vanilla
zest of 3 limes
3 eggs
4 dl flour
2.5 dl dried flaked coconut + 3 tablespoons for pan
2 teaspoons baking powder
1 teapoon salt
2 tablespoons rum
Lime Syrup
juice of 3.5 limes
1/2 dl sugar
Frosting
100 grams butter, at room temperature
100 grams cream cheese
3 dl powdered sugar
2 teaspoons rum
zest of one lime, plus juice of half a lime

Turn on the oven to 175C (325F). Prepare your cake pan by rubbing it with butter. Throw in 3 tablespoons of shredded coconut and shake the pan so that the coconut evenly covers the bottom and side. This will both help the cake from sticking to the pan and create a nice crust. Set aside pan.

While the cake is baking, make the lime syrup. Take the juice of 3.5 limes and add 1/2 dl sugar into a small pot. Put on the stove and bring the lime juice to boiling. Take the pot off the stove and set it aside to cool.
When the cake is out of the oven, take a fork or toothpick and poke holes all over the top. Spoon over the lime syrup, trying to cover the cake evenly. Set the cake aside to cool.
When the cake is cool, make the frosting. Whip the cream cheese and butter together in a bowl. Add in about a third of the powdered sugar, the rum, and about half of the lime juice. Whip together and add in the rest of the sugar. The frosting should lighten in color and get very fluffy. If you wish the frosting to be stiff (if you want to pipe it onto the cake and have it hold its shape), then you will need to add more sugar. I like it looser and I just spoon it on top of the cake. Adjust the lime juice and to your liking, remembering that the more liquid you add, the looser the consistency of the frosting.

Store leftover cake in the refrigerator.