The battery does not need to be there to start the computer.
Just pull the main battery , use only the AC supply, and it should free up the AC or wall supply to allow the computer to boot.
If not, you need more than a battery. Like maybe the dead battery took out the fuse in the wall supply. (You'll hurt yourself trying to get the Power supply apart, just buy new.)
Simple and cheap buy from E-Bay will fix both issues, Neither the battery or AC supply needs to be as expensive as factory or stoes want to sell them to be good.
Just find a well reviewed dealer on E-Bay for both items. You can run a laptop forever on a wall supply only with zero damage, thats how I've been running one devoted to a task laptop for 2 yrs.
Buy a battery if you want one from someone who has sold hundreds of them with a greater than 90% rating and save a ton of money. Same with the wall supply, just make sure it has the same Voltage and Current ratings, again, save a ton, and get a very good product.
Questions? I'll be around. Oblong really.