I had the same problems. Here's what worked for me.
Try using the factory charger brick with the USB cord that you use for your phone. I find the factory cable that came with my T110 charges much more slowly (sometimes won't bring the battery to a full charge overnight!). Then ... patience, amigo ... you
may as well wait a few hours til the little charging light goes white and you've got a full battery.
When you've got a full charge, press and hold the power button til the little light near the center of the top of the screen bezel (the black band surrounding the screen) blinks. It should come on then. (Same thing to turn it off - press and hold the power
button til the light blinks.)
Update to Bios v. 304. It isn't perfect (the battery still drains when the tablet is in use plugged in!!), but other than that it seems to be okay. In particular, it seems to solve the "won't power on" issue.
I used to be a real ASUS fanboi, but let me tell you, the power issues with this machine have really taken the shine off the brand for me.