What's the best way to test an LED
If it's a plain old red, amber/yellow, or green LED, many DMMs' diode test functions are capable of lighting the LED safely. Chances are decent that you just have it soldered in backwards, and the resistor
may have protected it from damage. (Unlike a normal diode, LEDs are not very good at blocking reverse voltage.)
If it's a more modern LED type, many DMMs can't develop enough voltage to light the LED, so they're not very helpful.
Lacking a DMM in your situation, or at least one capable of doing the test, I think I'd just desolder the LED's hookup wires from the PCB and touch it to a 9V battery both directions until I found out which side was positive. If neither direction lit it up, I'd call it dead and move on.
Going forward, be sure to pay attention to which leg is the short one by consulting the LED's datasheet. There's a convention, but it isn't binding; some LEDs are "backwards." You can also learn to study the LED's inner construction through the lens to determine which terminal is the anode.
I'm using an OPA2132PA so I think it should be fine?
That's the point of the voltage tests in the op-amp notes article. Notice how close to 9V you get before you start clipping with that op-amp with headphones requiring higher voltage. That means you get very little use out of the battery, as pointed out by another of the articles I pointed you to. A third article I pointed you to brought up the issue of virtual ground shift, which eats into this still further.
More voltage means you drain both batteries more fully, making for a more efficient whole.
I still have a lot to learn...
That's the thing: the circuit is simple, at one level, but fully understanding it requires that you grasp a large swath of the field of electronics. It's a good teaching tool because you aren't required to grasp the whole swath at once, however.