Kodak EasyShare ONE no longer comes on.

This is my daughter's camera. EasyShare One is the proper model name/number for it. She left it unused for a while,after having it behave normally for a year or so. Now, it refuses to come on. This camera has a 3" touch screen that resides on a thin metal plate which swivels out from the camera body. This "opening like a book" action used to make the LCD screen come on. There is also a power button on top of the body. Opening the screen results in no display. Pressing the power button causes a light that sits behind the slide switch on the edge of the LCD assembly to flash at a rate of about 2 to 3 times per second for about 5 or 6 seconds-then-nothing. This slide switch is the one that changes the camera from picture taking to picture displaying mode.We have two batteries. We charged both of them overnight, trying to eliminate weak battery as probable cause for failure. Nothing changed.
We do have the USB cable, but since the camera won't come on, it is pretty useless. The camera has an electric socket on it's base, but we don't have the external power supply that I imagine fits in there. This camera was bought used. I'm hoping that there is a secret sequence of button pressing that will bring this camera to life.