If the LED does not illuminate, then you have a hardware failure that could be as simple as needing to take the battery out and run without.
All unibody Mac with built-in batteries have very delicate cells, so use great care to avoid puncturing them - even when discharged, you could start a fire.
You could take it in for service or try a DIY repair, and as long as you don't leave any small debris between the battery and the case (and ensure nothing is left inside the case that fits that description), the skilled DIYer can easily exchange the battery.
Depending on your budget, you might delay buying a battery and have it serviced to see if there isn't a different failure of the charging system or the computer itself that caused a dead battery or will prevent operation even with a good battery.
I'd take it in for service unless you have a high interest in learning or there really are no good options where you could mail or take it directly. Apple should be able to quote you a cost to repair the battery in a mail in service and you can decide from there whether to seek other quotes or have them work on it with the expectation that you'll pay for a battery or get a call if the repair costs more than quoted. At that point you can decide to go ahead or not, but if it's just a battery, you'll end up paying for the service and get a 90 day warranty on the machine and the work.