I have a 6 year old Asus K53SD laptop. Suddenly, yesterday the laptop won't start up. So, I removed battery and tried starting up on AC power only. It worked fine. Then, while on power and on running system, I tried to install battery back. And when the battery attached, the laptop shut down instantaneously. I don't understand what maybe the problem. As for the battery, there was no issue AT ALL (replaced last year).
Battery by definition is an array of cells. If any one it those cells fail, battery either dies or works in a faulty manner. Your laptop may detect this problem in POST stage and deliberately abort start up on battery power to safe guard your motherboard. Note that you can never really say "battery is alright" without having it tested by a professional.
TL;DR you may need a new battery