Actually, you don't even need to measure the voltage drop across the led terminals.
Suppose the supply voltage is 5.0V, you add a 330R series resistor and the led is red, so it's forward voltage is around 1.8V.
If the current is 9.7mA, then the total series resistance is (5.0-1.8)/0.0097 = 330. In that case, there is no internal series resistor.
If the current is 5.8mA, then the total resistance is (5.0-1.8)/0.0058 = 550. So the internal series resistor must be 550-330 = 220R.