I'd strongly recommend you consider a professional data recovery company for assistance with the files, if they are so important to you. Any DIY attempts could potentially do more harm, than good. I'd second what @hang-the-9 mentioned and check how the HDD is detected on another computer internally via a different SATA cable and externally through an enclosure or a docking station.
You might also want to try swapping the SATA cable and the port where the HDD is connected to on your PC as well and see if that will make the HDD recognized in Windows.
Be careful because any additional stress on the drive could be fatal.
No way to do it if a computer can't detect it. Have you tried reading it from another computer or a USB enclosure? Maybe the board is bad, at times replacing the board with the exact model and firmware version can help, but it's hard to track down a spare board, and it could easily either not work with your drive anyway or not be where the issue is.