Thanks to all for the replies. the Youtube videos were especially useful.
I'm not sure what to do now. The screen is still absolutely fine when switched on, so perhaps I should leave it as it is, although I've started lifting the outer layer in one small corner. What if I just peel off the outer layer and leave it like that? Or maybe I'll buy one of those glass screen protectors off fleabay.
Either way, I agree with Peterm1 and I've lost faith in the camera so I've decided to sell it.
Incidentally, I bought both the NEX 5 and an Olympus Pen E-PL1 on the strength of an Olympus Pen S which I bought second hand in 1966 for £12 and which I still have. I was hoping that these cameras might be the modern equivalent. That Pen S was superb, and still is, with a build quality which modern cameras can't match. I bought it for colour slides because it gave me 72 shots on a roll of film rather than 36 so the running costs were halved, but the lens was so good that I found the results were at least as good as my full frame 35mm camera. I've recently bought two Pen FTs as well but haven't tried them yet.