IPhone only works when on wall charger? (details)?

About 4 months ago, I dropped my iPhone 4 in water, and it died. It wouldn't turn on, even after days in rice. I put it on the charger for a full week, but nothing happened. Recently, I saw the corpse of my iPhone in a drawer and just stuck it on a charger. To my delight, it turned on and seemed to be working. I typed in my passcode, and came to a screen which told me to connect my phone to iTunes so that I could activate it. As soon as I unplugged the phone (which was at 97% battery), it died. I figured, it would revive once I connected it to my computer, but it didn't. Why does it turn on when connected to the wall, but not my computer? Do you have any idea of how to get my phone to turn on when attached to my computer? Also, is this a battery problem? If so, how should I go about replacing it?

The water shorted out the battery and it needs to be replaced.

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