Heat oven to 325°. Grease bottom and sides of an 8" x 8" metal baking pan with butter and line bottom of pan with parchment paper. Grease parchment paper with butter and dust paper and sides of pan with a little cake flour. Heat 8 tablespoons butter, golden syrup, brown sugar, and marmalade in a 2-qt. saucepan over medium-high heat. Cook, stirring often, until syrup thins and sugar dissolves, about 5 minutes; let cool for 10 minutes. Whisk in the milk and eggs; set syrup mixture aside.
In a large bowl, whisk together cake flour, whole wheat flour, baking powder, cinnamon, ginger, baking soda, and salt. Add reserved syrup mixture and whisk until just combined. Pour batter into reserved baking pan. Bake until a toothpick inserted into center of cake comes out clean, about 50 minutes.
Transfer cake to a rack and let cool for 10 minutes. Meanwhile, heat remaining butter along with sugar and lemon juice in a 1-qt. saucepan over medium heat and cook, stirring often, until sugar dissolves, about 3 minutes. Using a pastry brush, brush half the lemon syrup over top of cake. Invert cake onto cooling rack, discard parchment, and brush the remaining lemon syrup on bottom and sides of cake. Invert the cake onto a serving stand and wrap in plastic wrap; let cool completely. The cake tastes better the day after baking and will keep for up to 4 days. To serve, cut cake into squares and sprinkle with confectioners' sugar.