The LED board that I used wasn't intended to control the servo motor as well, but Francisco showed me how to get power and signal to the motor by soldering the power and ground wires straight to VCC and GROUND on the board. I used a pin on the ATTINY44 that was not designated for use.
This board is connected via 2x2 pin to the LED and button board. It is used to manipulate the 6 modes that the button cycles through.