How to derive an equation for the delay time to LED shutoff?
I'd have said your formula would be:

dt = (Ct * (Vbatt - Vth)) / IL

This gives a time value of 15.88 seconds

The circuit starts with a Vgs = Vbatt --> 5V and will discharge linearly from there to the 1.824 V, the FET Vgs(threshold) value.
Note that the LED may appear off long before this point is reached, since at very low currents there may be no visible light emitted to the human eye.

If you set up an experiment with real components, who knows what time value you would actually measure. The capacitor may be =/-10 - 20% and the diode reverse current would be highly variable too.
Even if the Vgs(threshold) was very accurate, as said earlier, the LED would appear off while there was still some current flowing, even if you used a sensor to measure the light output. 

In an experimental configuration it might be worth adding a high value resistor from Drain to Vbatt (perhaps 100 kOhm) and use an oscilloscope to measure when the V(Drain) rises to Vbatt - LED forward voltage level.