It's relatively uncommon on an eee, but the symptoms sound like a broken solder joint on the power connector. Happens all the time on certain popular laptops; I've just never heard of it on an eee. What happens is that there is enough connection for the laptop to know that the adapter is plugged in, but not enough to actually charge the battery. The symptoms you describe match what is often seen if the power connector isn't totally broken off.
But since you did get one flash of green, you might try keeping it plugged in for 8 hrs & then try again; it could just be taking a very long time to charge up.
FWIW, make sure you have no external USB devices plugged in. An eee that I owned a while back would essentially short out if anything was plugged into the left-side USB connectors ... a known issue, annoying as all get-out. I'd forget, be going along & plug in a stick, and that annoying whooshy-whee sound would accompany it going abruptly dead.