The console stays on for a second and then turns off

According to the
Support Page your power supply may need to be changed:

Flashing orange, solid red, or no light: Your power supply may need to be replaced. Try the following solutions:

1. Try plugging your console’s power supply into a different electrical outlet. Do not use a surge protector.
   
2. If the power supply light is solid red or flashing orange, unplug the power supply and let it cool for 30 minutes. Then, plug the power supply back into the outlet.
   

If your power supply light is still flashing orange, solid red, or unlit, your power supply needs to be replaced. You can
Order a replacement Xbox 360 power supply from our Xbox Online Service Center. Console registration is required to replace the power supply under warranty (the power supply warranty
is the same as your console warranty).

I mean the light of current power supply. I have a xbox 360 slim

Hi Nathan,

Thanks for your post, if the controller is staying on after the console is powering off, try hard resetting the console with these steps:

1. Hold the power button on the console for 10 seconds (while it is on)
2. Release the power button after 10 seconds
3. Press the power button as normal to turn the Xbox back on
4. Attempt to power the console off as normal to check if the controller continues this issue

Normally if a controller stays on after the console is powered off the controller will time-out and turn off on its own, so it should not cause any harm.

I hope this information helps, thanks again.

If your console is in instant on mode the power brick will stay on as the console is still technically turned on just in a low power mode, if you want to turn off the power brick you will have to completely shut down your console, this can be done by holding
the power button for 5 seconds

EDIT: I just conducted a second test. I had it so everything was connected, apart from to the console. The light was amber. I then made sure the console was off, and then plugged the brick into the console. It stayed amber. I then used the controller to
turn on the console and the light went straight off and the Xbox wouldn't turn on. This suggests it's the connection to the Xbox, or something inside. Opinions on this?