The fan is ether dying or has something lodged in it. The fan is located next to the power supply, not where your video was showing. Be very careful working next to the power supply as the risk of electrical shock is high because the leads are exposed. Follow the guide here:
https://www.ifixit.com/Guide/Apple+Thunderbolt+Display+Fan+Replacement/41648.
My recommendation is to first unplug, take the LCD assembly off, and disconnect the cables from the fan running over to the logic board. With the display connected to a thunderbolt equipped computer plug power back in (it will still work without the LCD) and see if the sound is still present. This just assures that the fan is indeed the culprit. Follow the fixit guide linked above if it is indeed the fan and remove the fan from the chassis and try giving it a good clean. If the noise persists, you'll need to order a replacement fan from eBay or something.