The diode equation did not give a good fit to the LED characteristics. The fit that I got for the smaller values of current indicated that a voltage, V0, needed to be subtracted from the measured voltage. The intermediate values fit better if a correction to the measured voltage, V, for the IR voltage drop across the milliammeter was made. The maximum IR voltage drop was 0.06 volts. The larger currents still tended to deviate from the modified diode equation.
I do not know why V0 is needed. It could be that light produces a contact electrification of the junction in the LED.