How to find out characteristics of a LED?
I'd suggest connecting the unknown LED together with current limiting resistor to the regulated voltage source and measuring its voltage-current characteristics.







simulate this circuit &ndash; Schematic created using CircuitLab

The voltage at which the "reasonable" current starts to flow may be considered the "on voltage" of the diode. The slope of the I-V curve  above this voltage will allow you to estimate the R2 value (if it is mounted in your diode).

$$R_2=\frac{\Delta U_{VM1}}{\Delta I_{AM1}}$$

Of course you should watch out to keep the current reasonably low to not overload your diode. (You can also control how bright light does it emit).
Unfortunately there is no method to state whet is the maximum allowed 
forward current for the unknown diode.