Ah, OK. To detect all keys with a matrix you just have to add a diode per key so the electricity can't flow back the "wrong way" through the matrix and confuse things. Then you loop over the rows powering up one at a time and reading off the columns. But if you don't want to solder...yeah, you'd have to add a breadboard or two, or get wire wrap tools or something.
I see you also don't have any debouncing code on the Arduino. Are you doing that on the Unity side, or just writing games where it doesn't matter?