Update. Maybe good news! I had my wife try starting the car while I pushed the contact on the k40k9 relay, the car started right up! I am not 100% sure now but I think that this may be the problem.

If I wasn't such a clutz soldering I would try resoldering the board. I may try it anyway. At this point I have nothing to lose, so I may try doing it before I buy a new one.