Solder long stranded wires onto one of the speakers so that it can reach all the way to the second jacket pocket.
Solder a JST battery connector to the back of the audio board.
Position your components on the jacket while flat on a surface, and plan where your triggers, speakers, and battery will go. Use jumper wires to extend the membrane keypad wires.
Use a seam ripper to carefully open seams inside the lining of the jacket to gain access to the pockets for the next step.