HDMI port not outputting. even when screen is disconnected

It sounds like the IO board may be failing. Check out this guide for instructions on how to replace it. If you think the adaptor you're using could be faulty, then I'd recommend trying to borrow a separate adaptor for testing, or taking the device into a repair store for testing. It might also be a good idea to inspect the interior of the device for other signs of damage—something like this could technically be caused by a fault on the main logic board as well. Good luck!