There are couple of thing you can try.
1. Take the battery out of the EEE PC and press the power button for couple of seconds to drain all the power from the computer. And then try to connect AC and then switch it on.
a.) Removing battery
b.) Unplug mains
c.) Press-and-hold the power button for 10 seconds
d.) Reinstall battery, plug in mains etc, then try again...
Follow the EEEUser ASUS EE PC Forum for more information.
Still if you do not have any luck. (In my case it did not)
2. Call the ASUS help line and you need to send the EEE PC to them and they will solve the problem. If your computer is under warranty the process will be 100% FREE. Also the service is quite good (in Australia).