Was she right to put it in the airing cupboard to dry?

I see no problem with that. It was put out of the way in a warm (but not hot) dry place.

Wouldn't the water have seeped up from the bottom of the laptop?

I guess that depends on how much water and if the water was contained, or allowed to run and spread out. There are feet on laptops to lift them up a little to allow air underneath. If the laptop were in a cookie sheet, or something with edges so the water was pooled and deep, then yes, it could get inside. But I am assuming the laptop was sitting on a table or some other flat surface.

I think it's the power supply then. By the way, iit was a even surface.

I think it's the power supply then.

And I think it is the battery because you said it does work if plugged in all night. But, with the sparks, who knows? It could be both, and/or the charging/regulator circuit inside the notebook. So I stick to my comment in #6 above.

The laptop now works fine, apart from:

1) Sometimes the screen goes blank on startup.
2) Windows says the battery is charging but the laptop only works on the mains.

I am getting a replacement battery tomorrow.

I'm not sure what to do about the blank screen randomly on startup (it very occassionally happens when Vista needs UAC permission to do something).
It is an Intel GMA4500 card (it won't let me change the resolution!), I have made sure Windows and the graphics card drivers are up to date. Graphics cards testers all come up fine, so I don't know what to do!
I have been told it may be the backlight.

Should I give it back to my client and ask him to use it for a bit and report back?

The blank screen may be the inverter going. Backlights usually go and stay gone. Next time, shine a flashlight at the screen and if you can see desktop. If you can, then it is probably the inverter.

But it would not make sense (to me anyway) to do it only when the UAC prompt appears. ???

Yes, I don't understand that either!

If the backlight was going, why would it do it randomly and not every time I boot it up?

Because it is going - not gone.