Try removing the battery and then power cycling the laptop. Without some kind of electronic diagnostic tool or thermal camera you won't be able to quite accurately tell what is going on sorry to say.
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I would check to make sure the wall adapter is the correct adapter for starters. It might seem obvious but a lot of people assume the barrel plug fitting is all that matters. It should be pretty easy to find the voltage and amperage you need.
Other than that I would try to open it up or research the model and common issues. A leaking capacitor for example can eat away traces on a board.
Also, use/borrow a multimeter and test the voltage output in the adaptor. Plenty of "dead" laptops I've seen were actually dead adaptors that failed to provide power and then the battery just ran out
I know it's the correct adapter, but I'll check the voltage out and see what it says. The laptop has been sitting in my garage for a while, but I turn it on every few months and never had an issue with it before. This time I hit the button and nothing.