Preheat the oven to 170C and grease a loaf tin with butter, lining it with baking paper.
Boil the kettle to make the coffee using instant granules or a machine, then let it cool slightly until lukewarm.
In a large mixing bowl, cream together the butter and sugars with an electric beater or wooden spoon until fluffy (about 3-4 minutes).
Beat in the eggs one at a time, mixing at a low speed between each addition, then pour in the coffee and milk.
Continue mixing the batter until it forms a loose coffee-coloured consistency.
Sift in the flour, cinnamon, baking powder, and salt, and fold in slowly until the mixture is thick and smooth.
Pour the coffee cake mixture into the tin and bake for 45-50 minutes, or until a skewer inserted into the centre comes out clean.
Leave the cake to cool in the tin for 10 minutes, then transfer to a wire rack to cool completely.
For the glaze, whisk together the coffee, melted butter, and sifted icing sugar in a small bowl until runny but thick enough to set when cooled.
Drizzle the glaze over the cold loaf cake and leave to harden before serving in slices.