Line the bottom and sides of an 8-inch square cake pan with two crisscrossed sheets of foil, leaving an overhang on each side.
Place a layer of ice cream sandwiches into the bottom of the dish, breaking them as necessary to cover as much of the pan as possible. Scoop the strawberry ice cream onto the sandwiches, then spread evenly with a spatula until the sandwich layer is covered. Add the remaining ice cream sandwiches on top to form another layer. Top with the cookies and cream ice cream and spread evenly. Spread the whipped topping over the top like a frosting, then top with the sprinkles. Freeze for 1 hour.
Remove the cake from the freezer, then carefully remove the cake from the dish and peel off any foil that has stuck to the bottom and sides. Slice and serve.