Electronic – I’d like the 12v rocker switch LED to illuminate when the switch is turn OFF

There are two possibilities the switch is inserted:

1. Switch switching the voltage line.
2. Switch switching the GND line.

^{simulate this circuit – Schematic created using CircuitLab}

The capacitors shown in the circuit are the capacities the (more or less) long lines form to GND.

If the 2nd circuit is what you have the lamp always is connected to alternating voltage. In that case there is a possibility of some very low alternating current flowing via C10 to GND even if the switch is open.

You can find out if your configuration is the 2nd circuit by testing with a one-contact neon test light. It is the case if the test light lights up when touching one of the two connections in your lamp socket even if the switch is turned off.

You are pretty much on track, resistor value is not really critical here, you can use 10K for both resistors, that will work fine.

^{simulate this circuit – Schematic created using CircuitLab}

You can make a thevenin equivalent circuit for the open and closed switch situations, for the closed switch:

^{simulate this circuit}

When you press the switch, you will be driving the base with around 1mA, that should make the transistor drive the 20K pullup low very comfortably (in this case, the transistor should be able to drive hfe*ib through the collector, which will be at least a hundred mA)

When you release the switch:

^{simulate this circuit}

You connect the base to ground through 10Kohm, ensuring you turn off the transistor. Quite simple and effective.