begin with a rather safe guess that an LED has a maximum forward current of 25mA (this covers most run-of-the-mill LEDs, some are higher, a few older types (mid 1970's vintage) are lower). build a current source that sources 15 to 20mA... better still make it adjustable from 5 to 20 or 25mA. if you build the variable one, you can calibrate the circuit with your multimeter (in current or milliamp mode), so you know where to set the potentiometer for various currents. then change back to reading voltage and insert the LED, and measure the forward voltage at various current settings. if you have a photometer, you can also measure luminosity at various current levels.
shown below is a variable current source for testing LEDs. there are two diode voltage drops applied to the base of Q1. one of those drops is equaled by the forward diode drop across the base-emitter junction of Q1, with the remaining diode drop across the emitter resistor (VR1+R2). the resistors provide negative (or degenerative) feedback when collector current is flowing, maintaining a nearly constant collector current. the collector current is a tiny bit less than 0.7/(VR1+R2). R2 provides a limit of current of about 30mA when VR1 is zero ohms, VR1 provides the ability to adjust the current from about 5mA to the 30mA limit. do not use tiny cermet pots for VR1, use a "normal" size one, typical of a balance control or a motor speed controller. it should also be linear taper.