Switched LED light on without ground (Jeep)

[SubmarineAZ]

I wired in an auxiliary fan to my Jeep radiator. Details:
1. Fan grounded to body.
2. Power wire for fan goes to standard 4 pin relay (with prewired harness)
3. Constant hot 12V for relay goes to 30amp breaker then to battery.
4. Engine-on power12V for relay goes to fusebox in engine compartment.
5. Ground wire for switch runs into dash, where it's connected to a standard 2 pin switch which is grounded to the body.
6. Fan as wired works fine, turns on and off with engine on, does not run with engine off.
7. Ground wire from switch to body generates solid tone on multimeter.
8. Ground wire from switch to relay generates no tone on multimeter.

SO I wanted to wire up a small LED light to remind me the fan was on when the switch connects to ground. Simple right?

1. Connect LED to constant 12v in the dash.
2. Connect LED to ground wire on switch-to-relay side which is not grounded.
3. LED light comes on!
4. Recheck wire for tone with multi-meter and any cuts or ground faults. Nothing!

I'm so confused.

 

[brianmoooore]

4. Engine on power. This is effectively ground when the engine is off, so if your Led supply is constant live, it's lighting via that and the relay coil.

 

[SubmarineAZ]

4. Engine on power. This is effectively ground when the engine is off, so if your Led supply is constant live, it's lighting via that and the relay coil.

With the key off, the LED does not light. I tried just touching the LED wires to the battery and it only works one way- red/power wire black/ground wire. Switching them won't allow the light to work. But if that weren't the case, how would a ground wire from the relay become a powered wire with the key off? Wouldn't that short out? I'll scan the manufacturers diagram with some added info about what I'm doing with the switch later. Used the fan all yesterday afternoon and everything worked fine sans light.