try putting in a separate relay for the starter solenoid as i had the very same problem,
i would try and start it and get no noise what so ever, i put a mulitmetre to the battery and read 8 volts
the corolla's have a tendancy for the key starter switch to become high resistant so the volts drop
its pretty simple to wire up, i just took the lead that goes to the ignition for the remote power and connected it to the switch side of the relay, then i had a new big lead from the battery to the relay and the same size cable from the relay to the starter solenoid. USE FUSES!!!
hope this helps because my corolla starts everytime no matter what state the battery is in.