Step 3: Adding some wires
Now you know what needs to be connected to what, solder some wires to those places. (colours of the wires doesn't matter)
Now try and control the car by touching the wire ends together, if it doesn't work, figure out why, it could be a simple bad solder joint.
Drill some small holes so the wires can get out, put the wires though them, take care making sure that the wires won't be bothering any of the mechanical parts inside the controller (such as the stick or the metal flap, or stopping the casing from shutting completely).
Close the case, test it again by connecting the wires and by using the sticks, it's better to have both of control methods working.
If you want to make the controller one big base-less joystick, cut a large dowel (or pipe), and attach it to the top or bottom of the case before you close the case. Make sure everything fits.
Below are three examples of what I soldered