Could be the battery is toast and the laptop will not start due to a non-functioning battery, some devices are that way, or it could be the motherboard.
I have to say I would be leaning towards the battery as the issue, since you get at least a small response when plugged in. You can try the following and see if resolves the problem.
Unplug your charger cord.
Remove the battery.
Press and hold the "Power" button for 30 seconds.
Now plug back in the power/charger cord.
Now put the battery back in.
If this doesn't help, then test the power cord to see if it is good. Either try your cord in another device, or try another devices cord in your laptop. You may also want to try charging it when the device is off, rather than on. If it works then, the cord is probably the issue.
Should that not be the problem, then check the battery to see if it needs replacing.
If you do not feel comfortable doing this yourself, then I would try a local tech.