Grab the input power coil (more square circuit board) and with the help of a 3rd hand tool, solder the red wire to the AC adapter's red wire, and likewise with the black wires.
Use a heat gun or lighter to shrink the smaller pieces of heat shrink over the solder joints, then the bigger piece over the whole connection.
Alternatively, you can plug your AC adapter into a terminal block connector and use a screwdriver to secure the circuit board's wires to it. This eliminates the need to solder, but only on this half of the circuit.