I raised this thread in Apps, if I remember correctly, and Jeffrey kindly pointed me towards the docking/charging/drain problem http://www.transformerforums.com/forum/asus-transformer-help/3891-asus-transformer-battery-drain-facts-real-solution.html
I've read through this and checked the 'wake-up' and it doesn't wake up when turned off. I have noticed that when I thought I had turned it off at night, in the morning it would start up when I lifted the lid, but not always.

I'm not sure if this battery drain problem affects the pad as well because if I connect power to the pad only I still get 'no charging' message in Settings/About. My rapidly failing computer skills are telling me that the fault may well be with the cable or transformer. Unlikely that both batteries (dock and pad) have both gone, so I'm going for the cable/mains connector. I did notice a couple of days ago that the plug felt pretty warm but now it is stone cold.

I've been on to the supplier (I've only had it 4 days) and arranged a replacement, but is there any way I can tell if I have the dock that causes this problem (serial number?) or is it just pot luck?