How do I find out about the technical details of an LED if I don't have the fact sheet?
Short answer: you can't. But you can do some smart non destructive trial and error.

First of all you need to at least know what current/voltage you are expecting: are they power leds, are they small 3mm leds, are they smd or not, what color are they? Try to find a datasheet of a led you think is similar.

Let's assume that you find a somewhat related datasheet so you have the forward current and voltage. You will need a bench psu for this, or a power adapter and a variable resistor, and in both cases a multimeter. You start by building a circuit that will let some 1/10 of the current from the datasheet at the same forward voltage, you then slowly increase the current until you are satisfied. You can possibly burn some leds but you'll probably get away with it.

Shorter answer: no datasheet, no buying.