A few ideas…
Is there actually any power going to the WD, i.e. does the HDD activity light work (on my own WD MyBook, there’s a funky blue LED on the front, indicating when the drive is powered on).
Is the drive still under warranty? Not sure what these drives are like inside, but if the warranty has expired it might be worth thinking about removing the drive from its casing and hooking it up internally to see if the casing and/or external power supply might be the culprit. As I’ve never done this, perhaps others can provide further guidance if you choose to go that route.
Sounds really silly this idea, but…you have tried the drive plugged into different electrical sockets? (Hey, you never know…;)).
Of course, a data recovery service is always an option if the data is critical, but this is expensive and I wouldn’t pursue that one unless I was really desperate.