It may be that the actual connector on the power button has failed with time (it is a 2009 camera..that’s a lot of button presses?) or there is something obstructing the connection (like spilled drink/dust). Try spraying compressed air to clear any debris, use a cottontip/toothbrush with isopropyl alcohol to clean any sticky stuff. Failing that you will need to remove the outer case (and quite a few other parts) to access the power button/board and change it.
This is quite a good video of the process https://www.youtube.com/watch?v=6hymECkCBvo to open the camera.
powerboard ($40-100+) is available on ebay as is the connector ($14).
Good luck