Preheat oven to 170°C and grease/line a 2lb or 900g loaf tin. I used a 8.5x3.5x3" loaf tin.
Grate the lime zest and rub into the castor sugar, set aside. Squeeze the lime over a sift and set aside juice - we will use later for the glazing.
Using hand whisk or electric mixer, beat together butter with sugar and lime zest mixture until light and fluffy.
Add in the eggs one at a time, ensuring that it is well incorporated before adding in the next egg.
Whisk in milk lightly, then gently fold in the desiccated coconut
Mix flour in until well combined, then pour cake batter into the prepared loaf tin.
Bake for 50-60 minutes, if it is browning too much, ensure to cover the top with aluminium foil towards the last 15-20 minutes. Check with a skewer inserted in the centre to ensure that it comes out clean.
While the cake is baking, mix together lime juice with the 2 tbsp caster sugar. Sugar should dissolve into the juice by the time your cake is ready.
When the cake is done, remove cake and place on cooling rack. Poke holes all over the top with a thin skewer or cake tester, then either drizzle the lime sugar syrup over it, or brush the syrup on top using a pastry brush.
You may wish to sprinkle some desiccated coconut on the top, or leave it as it is. Let cake cool completely on cooling rack.
Store leftovers in an airtight container and eat within 2-3 days. If wrapped well, this cake can be frozen for up to 2-3 months. Defrost and warm up in microwave before serving.