I tested the cheap microphone when it arrived, and sure enough it had a horrible high frequency hiss that was something like -30 or -40 dB. It was so bad that I initially thought it was picking up the sound of the AC, but I moved into a closet and blocked the entrance with a blanket, and still got the hiss.
I proceeded to gut it and set about replacing the internal parts. It took a few tries to get it right, which is a story of its own, but I finally got it working.
Listen to a quick sound test:

Here are some pictures of the build:
This is a picture of the circuit installed in the microphone body, along with the new capsule in a mount that my brother made with his 3d printer:
This is the microphone with the cover replaced: