It will probably work. Even though many "better" ways are possible, if this is what you want then you might as well give it a try and experiment. If you get really lucky, maybe changes in the static air pressure will be visible in the LED light.
One minor caution would be the make sure you connect the diode the correct direction. While it's just a diode with higher forward voltage drop, do keep in mind all diodes also have a maximum reverse voltage and a certain amount of reverse "leakage" current before hitting that breakdown voltage. No diode perfectly blocks everything when connected in reverse. With normal (non-zener) diodes it's usually dozens or even hundred of volts, and leakage is usually nanoamps for silicon or microamps for silicon-metal (Schottky) diodes. With a high power LED, it's *probably* more than 12V breakdown and likely to be not more than than milliamps leakage, but no effort has been put into the LED's design to give you a useful breakdown voltage or reverse leakage spec. If you accidentally connect the LED backwards, these specs could turn out to be surprising.
If using a PC power supply, also keep in mind modern PCs can deliver several hundred of watts at 12V. Best to first experiment with a lab bench power supply with the current limit set to no more than you know you need, or at least something like a "wall wart" power adapter which is limited to a lot less power.