The attached file is just the schematics for the L100 series. Print out pages 218 and 219. You are troubleshooting two separate problems. First is the ignition circuit and the other is the starter circuit. They are not related.
To get the starter spinning you need the PTO switch OFF, the brake switch fully depressed and the key in the start position. You can follow that circuit from the S terminal on the S1 Key Switch down to the S terminal on the solenoid. Now, if the brake is depressed or locked in the parking brake position, then perhaps the brake switch has popped off of its mounting tab. This happens occasionally and you have to snap it back in position. You can see the brake switch just inside the pedestal on the left side.
The other circuit you're troubleshooting is the ignition. For you to get spark, you need the magneto line (start at X9 on the right side, just above the magnetos) to be ungrounded. If it is grounded by any switch in its path (S1 key switch, S2 PTO switch, S3 Brake switch, K2 RIO latch relay, S5 Seat Switch or S6 RIO switch) then it may turn over, but it won't start.
Something is bothering me. You said that you disconnected the seat switch and it started. In a properly configured L130, that shouldn't happen. When the seat switch is disconnected a spring loaded shorting bar in the plug moves down and shorts wires 951 and 130 together providing a ground to the mags. IT JUST SHOULDN'T START!
I'll let you ponder the rest of the wiring, but I'd first go out and make sure the brake switch is mounted correctly. Here is a shot showing where it's mounted:
And here is a closeup showing the rubberlike mounting ears that sometimes pop out of that slot in the bracket:
Good luck,
Paul
EDIT: TobyU posted while I was preparing my post, but we essentially said the same thing. CHECK THE BRAKE SWITCH.