Sounds like your charging cable is toast or the power jack inside has failed or been damaged. It's not uncommon for the solder points to break where on the power jack inside. The power jack is the part of the unit that the charging cable plugs into. There are wires that attach to it and lead off to the motherboard and battery compartment.
If the unit fires up for a few seconds with the battery installed but NOT with only the charging cable installed, then it's likely the battery is nearly dead from not being charged due to a bad cable or power jack, but not entirely dead, so it does try to start. Since removing the battery causes it to do nothing, we can assume that no power is getting to either the battery or the motherboard through the charging cable or power jack. One of the two is bad almost surely.
I'd get a new AC charging cable, you can find them for under ten bucks for most models, and if that doesn't cure the issue, either replace the power jack or have an electronics or computer repair service do it. The power jack is normally less than twenty bucks for the part and probably (Since repair shops are so proud of themselves) about a hundred bucks to have replaced. It could be less. I'd call around to see what you're quoted. You can find disassembly tutorials by googling your model number and probably do the repair yourself.