You can still try a different USB cable even if at first everything seems to be ok with the one you are using. Maybe it's losing power somehow and doesn't have enough juice to power on the display.

If it's not that, I guess your Trezor's screen died or is dying on you. A Ledger Nano S screen is easy to replace but I don't think you will have much luck with the Trezor One.
There is a trick you can try with your phone camera. Connect the Trezor and fire up Trezor Suite where it asks to enter the PIN. Then, turn on the camera on your phone and point it at the screen of your Trezor. Look at the image on your phone's display and check if you can see the PIN pattern. Adjust the angles. Try both with the front and back camera of your phone. Experiment with different lights in your room. More brightness, less brightness, etc. Maybe it will work.

Some say you can also try pointing a flashlight at the display or turn on the flash on your phone. 
I would also contact Trezor support and tell them what is happening. Ask them for advice.

Did you maybe drop the device on the floor? Can humidity be a reason the Trezor is dying?