Failsafe mode Won't crank over. Tried all the keys. Lock light is on.

Connect it to a professional grade scan tool computer.

Sometimes the engine failsafe mode is triggered due to a sensor failure or an electrical glitch. It's also common for a security system fault to engage this warning light, which means it's not always a mechanical issue that's causing the failsafe.
Oct 31, 2018

What is the Engine Failsafe Mode? ' YourMechanic Advice

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Vehicle Engine Failsafe Mode Explained and How to Reset it

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In most cases, the check engine light will turn ON as the engine switches to failsafe mode. But having a check engine light does not ...
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‎What is the engine failsafe... · ‎Fail-Safe mode isn't always...

With a voltage meter see how many volts are at the starter terminal. If not 12+ volts you have a bad connection. Check connection at the starter solenoid. Does it have 12+volts? Short the solenoid. Does it put voltage to the starter?

you have a shorted battery (have it tested) or there is a loose connection. A starter motor pulls a large amount of current when it engages, a loose conection or shorted battery will all the voltage away from the dash,lights etc.

Are you saying the engine cranks but won't start, or the engine won't crank ?
If the engine does not crank, you would check for voltage at the starter, the neutral safety switch, and the key switch as well as the ground connections.

Is the battery dead? Check your battery condition and connections. Check your starter fuse from there. Check the connection to the starter then progress to the ignition side of the concern

Check engine light on , failsafe mode . The first thing to do would be to hook up a scan tool an check for DTC'S - diagnostic trouble code's . Was the vehicle overheating ? Failsafe mode will shut the vehicle off to save the engine if it is over heating . Do you know if the vehicle has electronic throttle control ? Was their a yellow wrench symbol lit on the instrument cluster as well ?

The ignition switch and ignition lock cylinder need replaced

Sounds as though when your battery died the engine imobiIiser locked down.
YOU MAY HAVE TO HAVE THE CAR TOWED TO A LEXUS DEALER AND HAVE THE MASTER KEY AND OTHERS REPROGRAMMED TO THE IGNITION...BUT
If you have your wallet key as well , try taking all to the airport. HOLD the card to the dash beside the ignition for at least 20 seconds, then try turning the key several times. I haven`t done an LS but the failsafe when a remote fails is to hold the key beside the ignition for at least 20 seconds ....
Also try the lock routine, but I think the issue is in dash..security needing to reboot to your key.
My understanding is that the lock routine re initialises the lock setstings when power is restored

try disconecting the battery for 5 mins see if it reeboots computer.

does the engine completely shut off? It sounds like the electronic throttlebody is causing the problem.