Preheat oven to 350°F (175°C). Line an 8x8-inch baking pan with parchment paper.
Make the crumb topping: In a medium bowl, whisk together the brown sugar, granulated sugar, flour, cinnamon, and salt. Pour in the melted butter and stir until large crumbs form. Fold in the sprinkles and set aside.
Make the cake batter: In a large bowl, cream the butter and sugar together until light and fluffy, about 2–3 minutes.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Add the sour cream and milk and mix until smooth.
Add the flour, baking powder, and salt. Mix until just combined, then gently fold in the sprinkles.
Spread the batter evenly into the prepared pan.
Sprinkle a small handful of the crumb topping over the batter, then evenly distribute the remaining crumbs on top. Gently press the crumbs into the batter slightly.
Bake for 34–40 minutes, or until the center is set and the crumb topping is lightly golden.
Allow the cake to cool completely before slicing.