Phone doesn't react – How can I find out what's wrong?

Hey everyone,

I was using an iPhone 3GS which I bought in January 2010. My power button hadn't been working for a while now but I could get around with Apple's native Assistive Touch. I realized over the last days that my battery had become very short-lived compare to before, and one day after I had just fully charged it, taking it out of my pockets it didn't react at all anymore. Plugging it in again there wasn't any reaction either.

How can I find out what happened to the phone? My wild guess is that the battery just stopped working, but any other suggestions are welcome. What should I do as a first step?

Many thanks for your help!