How to calculate for how long an LED emits light when in series with a capacitor?
I think simulation is going to be the best approach here. Please read these questions which have information about simulating LED's using the Schockley diode equation.
LED modeling with Shockley diode equation
How do I model an LED with SPICE?
The circuit editor on this website also has a simulator. I have added below a schematic that can be a starting point for the simulation you want to do. You will want to do a time domain or transient analysis. There are other free simulators (such as LTSpice).
I have used the default LED component. But if you open the circuit up in the circuit editor, you can change the LED parameters and then run the simulation. You seem pretty smart and I am hoping this is enough information for you. To go much farther I would have to just about take on the problem 100 percent myself and I hesitate to do that because of the time investment.



simulate this circuit &ndash; Schematic created using CircuitLab
V1 is a step voltage. I edited it to set the step to 9V. This means that at time of 0 seconds, the voltage transitions from 0 to 9V. This is akin to your circuit where you said that the capacitor was initially discharged.