Check that the power button is working OK by trying an SMC reset:
Leave power on with just battery for 30 minutes or more, so if you plug in the power adapter, the LCD on the plug changes to amber, indicating charging.
press and hold the left-side shift-Control-Option, then press and release the power button. The adapter LED should instantly change to green, then back to amber after a few seconds - which indicates a complete reset. If the adapter goes to green while you hold shift-Control-Option, before you even press the power button, then, the power button is sticking. You COULD try gently cleaning underneath and around the edges of the button, but I haven't seen one fixed doing that.
Also, try disconnecting both the power supply, and the battery for 30 seconds, which is an alternate method to reset the SMC. It MIGHT fix your power button issue.
If you can't get any better results, I suggest taking to an authorized service shop, at least to verify what you have. I doubt that you can do anything except replace the top case.