Electronic – LED burning out due to high current

1. I've used 220 Ohm resistors in the past for all of my LEDs (at least that I can remember), however nothing on my arduino+breadboard with a few leds on the 5V breadboard rail had worked.
    
   Blindly using a value based on lore or superstition instead of math is a bad idea.
2. I finally looked at the data sheet
    
   Wow, what a radical idea! Just imagine, now the people that design a part do the careful research, find out its parameters, and publish them. We can read their results and recommendations and use mathematical calculations to decide how best to use the part and have a good idea up front what it will do. No more waving of dead fish during a full moon. This changes everything!!
3. This explained perfectly why they had not started working
    
   No, it doesn't. The maximum current rating is just that, the maximum the LED can take. The light output of a LED is pretty much proportional to the current thru it. While your LEDs could have been driven with 35mA, they would certainly have lit up noticably at half that or a quarter of that unless perhaps you were viewing them in direct sunlight. Humans perceive light intensity logarithmically, not linearly. That means each halving of light level looks like a fixed increment to us. For example, the sequence of 100%, 50%, 25%, and 12.5% of max brightness will look like roughly even steps towards dark to us.
4. So was the resistor really hot or cold? I've touched a few hot parts in my career, including getting burned a few times, and it's never felt cold. Really hot feels hot, and you have a reflex reaction to pull your hand back before the brain engages in concious thought. This concept of overloading the nerve endings makes no sense, and makes me think it really was cold and you just somehow convinced yourself it was hot.
5. You are right, a 95Ω resistor with 3V accross it will dissipate about 100 mW. That is enough to make a 0805 obviously hot, but shouldn't burn your finger. A 1/4 Watt thru hole resistor should be noticalby warm but not so hot you can't hold it for a while.

Something is not as you think, but it's not clear what. Have you tried the LED in both orientations? It sounds like it is hooked up backwards and therefore not lighting.

I would use a 300Ω or so resistor with your 5V supply and try several different LEDs. Just about any LED you are likely to get can take 20mA, so the 300Ω will limit the current to well within the safe operating range while still allowing for plenty of brightness indoors on your bench. If in doubt, get another known working LED from someplace and figure out how to make it light. Then substitute your LEDs trying both orientations. If they don't light, then they are blown.

Update:

Op is using a keyboard controller that apparently has a built-in step-up switching regulator IC or circuit for a constant current LED setup, with no idea of the specs, target current, voltage, or wiring for the LEDs. Considering the Logitech K800 is stated to have a 10 day life on fully charged NiMH with the backlight on, there's PLAINLY a short or something on the board or wiring. Check the original board for wiring details, and double check your modifications.

Old Answer: _{There is no way for 2x AA NiMH Rechargeables to be 15V. The nominal voltage will be 2.4V. From your description, you're obviously guessing the led voltage based on their typical recommended max, which is not how it works at a lower voltage. For example, a typical blue led will only be 3.3V Forward Voltage at 20mA. At 1mA, it may be down to 2V.}

_{As for measuring it, the voltage across a circuit will always be the voltage across it's voltage source. And the current through a circuit is equal throughout the circuit.}

^{simulate this circuit – Schematic created using CircuitLab}

_{Notice, these 3 leds, with a total Forward Voltage of 6.3V at 20mA, will only have 0.85V or less across them, and 9 microamps. That's 0.000009 amps or 0.009 milliamps. The total voltages add up to the 2.4V across the batteries.}