Is it possible to approximate specs of a LED?
As indicated in the comments it is easy to make some basic assumptions and be fairly safe. Just about any 3 mm or 5 mm LED will have a max current of 20 mA with very little benefit in terms of brightness once you get to around 10 mA.

Do you have a multimeter handy? If so follow this process:

Based on your chosen battery voltage pick a resistor that will give you a current of 5 mA. e.g. for 3 V use 3/0.005 = 600 ohms. Any value within 50% of the calculated value will be OK. 

Connect the LED into the circuit and measure the voltage across it. Subtract that from your battery voltage and recalculate the resistor values needed 5 mA ( (3 - x)/0.005). Aim for a resistor between the calculated value and half of that number to put you in the 5-10 mA region and it should work fine. 

Alternatively if you don't have a meter but do have the Wii console and controller on hand the put the wii into the sensor bar calibration mode which shows you the controllers sensor reading. Start with a large resistor (e.g. the first value calculated above) and slowly decrease the resistor until the LED is clearly shown in the sensor readings.


  Or is reusing parts (without available specs) generally considered bad
  practice? I don't want LEDs to explode or catch on fire.


It's bad practice for anything requiring any level of accuracy or control. Or anything that will be resold and so needs a reasonable level of quality control. But for something like driving an LED of unknown type for a home project it's perfectly reasonable to do. LEDs have a huge operating region so you can afford to be a little sloppy in how you handle them.

There's no danger of fire or explosion, worst case they make a tiny pop sound and a small puff of smoke.