Dropped my iPhone 5s in the bath?

Dropped my iPhone 5s in the bath yesterday afternoon and it was still working fine so I carried on using it. Last night I switched it off, but it kept turning itself on and siri came on about three times. I also noticed that the touch ID wasn't working and I could see water behind the screen. When I woke up this morning it wasn't working even though I'd left it on charge all night (and the charging was working). Now the iPhone's telling me to connect it to iTunes but I've tried to restore it twice on iTunes, which is what it's telling me to do but it won't work. Now I've taken the sim card out and put the iPhone in the airing cupboard.

Is it a lost cause? Am I doing the right thing?

You should have turned it off immediately and let it dry out for a few days before trying to turn it on.
Now it is likely that short circuits have caused internal damage.

You can try to turn it on after letting it dry for some time, but do not expect miracles.

Of course it is not. The moment you get water into an electronic device you need to remove all power including the battery. Then thoroughly dry it out which can take 4 to 5 days to allow water in unreachable areas to evaporate. Otherwise the water starts an electro-chemical reaction which eats into components and the board itself. So even if you do recover it now, it is likely to never work correctly and those reactions continue.