Why can't I turn my computer on?

Start at the beginning >> the brick, and verify there is the specified voltage on the connector. When the brick is plugged in, it should run the computer even if the battery is bad or not installed. If it does not, then there is an issue with the socket on the computer or the power button circuit. The former being much easier to fix than the latter, but it must be opened up. You might not want to attempt that, and professional help will probably be economically not warranted.