The battery is unlikely to be the problem. All laptops I know work fine without the battery. Some laptops may have trouble running with a defective battery - the battery may claim to be ok but fail to deliver enough power - but even that is unlikely.
Most probably, either the power supply is broken, or the laptop itself (probably something on the main board, or maybe even the power button ;-)).
You can try to check the power supply, by either switching in a known good one (if you can get one, e.g. from a friend or colleague), or by measuring its voltage (after looking up the expected voltage). If that is not it, the laptop is defective and has to be repaired.
The difficulty of repairing a laptop ranges from easy to almost impossible, this depends very much on damage and model. At any rate, unless you have experience with electronics repairs, it's unlikely you'll be able to do anything. Try finding a small shop that will give you a quote for a repair at a low price - or just sell the laptop as defective on ebay or similar, you may still get a few dozen $ or € for it.