So it DOES boot if you disconnect the battery and the charger? You're essentially resetting the embedded controller each time you do that, so it sounds like there is a problem with it. You would need to contact the manufacturer or reseller you bought from if it's under warranty; that way they can service it at no cost to you (maybe just shipping cost).
If it's not under warranty, a local repair shop should be able to fix it, but unless they have good soldering skills and can solder a new EC chip for you, they'll probably replace the board - which won't be cheap.