Assemble the bits in the following order: Battery + Cable > Power > Button > Wire > LED.
Mount the Power/Button combo to one of the wide sides of the Battery so we have a solid surface to press against. Use a folded-over piece of duct tape between the bits and the Battery to stick them together.
Use small pieces of duct tape to mount the Lego brick (or similar object) on the Button bit to create a better actuator. This will make it easier to press the button.
Mount the LED (connected to the Power/Button assembly via the Wire) to the bottom of the battery, using doubled-over duct tape.
Slit the cardboard tube lengthwise to open it up and expose the inner surface.
Set the whole assembly down on the tube so that the LED sits just inside one end and the Lego button sits somewhere in the middle. Mark the location of the button, remove the assembly, and cut a hole in the tube for the button to stick through.
Now put the components back, with the button sticking through and the LED again just inside the end of the tube. Use duct tape to secure the whole assembly to the inner wall of the tube.
Wrap the tube tightly around the assembly, then wrap duct tape around the tube (leaving the button exposed) to hold everything together.
Cut out a small square of paper towel, a little bigger than the end hole of your tube. Place it over the end containing the LED to create a nice light diffuser, and use the rubber band to secure it in place.