How to tell if this light's bulb is blown? And what kind of light is it? See photo. Is this a normal LED? I'm trying to troubleshoot it and can't tell if the bulb\diode is fine.
It's an LED.

Easiest way to see if it is damaged is to connect it to a battery and see if it lights up. 

The difficulty is that you can't disassemble the the white plastic unit, so if the LED does not light, the connection  wires, resistor or LED itself may be damaged.

If you have a voltmeter you could measure the resistance across the terminals. It should be infinite one way, tens of ohms the other. If it's infinite both ways or zero either way it's trash.

Good news is that decent quality rechargeable LED lights are now inexpensive and ubiquitous.