I may as well ask a question or two of RK and throw my opinion in.
Before you took anything apart and started jumping wires, what exactly did the machine do when you turned the key to try and start the engine? Did it do absolutely nothing as if someone had removed the battery?
I ask this because before you dive in too deep with wiring diagrams and such, a couple of very simple voltage tests might help determine if the issue is an engine issue or an equipment issue. Actually reading my post will take longer than the actual testing.
It looks like you may have already done one of the tests. If that red wire you unfortunately jumped to the white "ground to kill" lead connected to the B terminal on the key switch then the question of, "do you have power getting to the key switch" looks to be a definite yes.
The second test I would do since you have easy access to the key switch is, plug the key switch back in. Assuming you didn't blow the fuse by shorting power to the white wire, and you still have power on that red wire, take your voltmeter set to DC volts of course and put the black lead of the meter on a known good ground. Now take the red lead and touch only the S terminal on the key switch once it is plugged back into it's connector. Once you find which wire connects to the S terminal it might be easier to gently probe it from the back side of the connector but don't pierce the wire itself. Now, turn the key to the start position? Do you now get voltage on that S terminal?
If have no power to the S terminal you more than likely need a new key switch. If you do now have voltage on that S terminal there is one more simple test. After that, it may be time for the wiring diagrams. Get the engine to crank first and then you can worry about if you damaged the ignition modules or not.
So, if in the previous test you did indeed get power to the S terminal of the key switch when you turned the key to the start position, the next test will be at the starter itself. Find the small spade terminal on the starter solenoid. If the start circuit goes through the Kohler wire harness that wire I call the excite wire will be blue. Sometimes the OEM of the equipment has the engine built so it runs off their wire directly from the key switch. If that's the case that wire will probably not be blue but either way it's the only single spade terminal on the solenoid.
This is where you may need another person to help you. Again attach the black lead of your voltmeter to a known good ground. Now take the red lead and while leaving that excite wire connected to the starter spade terminal, touch the exposed portion of the spade and have your helper while sitting in the seat, turn the key to the start position. Do you read battery voltage at that spade terminal now? If you do have battery voltage at that spade terminal you probably have a ground issue.
If only it were that simple right?