My Toshiba external 500GB ended up the scrap bin last month with exactly the same symptoms as yours. I couldn't test it because Windows diagnostic software couldn't detect it either (not surprisingly, since Windows couldn't).
Because I'm aware that external hard drives are considerably less reliable than internal ones, I have for several years backed up my data to four separate drives, so even if two should fail together I don't lose any data.
I'm afraid I cannot give you any good news about your drive. Most times it's actually the USB adapter board inside the enclosure that fails, not the drive itself. Extracting the drive and putting it into a third-party enclosure is the usual fix when that happens, except the Toshiba external drives have a proprietary interface - it's not the regular PATA or SATA interface (or at least mine wasn't). That means it can't be used in a different enclosure.