Firstly as I'm sure you are aware, electronics without a surge protector is a bad idea.
I guess firstly i would check the power brick, what color is the light (when Xbox is turned on)?
Green means it should be OK
Orange or Red means you have an issue.
Light is orange/yellow - It is not necessarily a problem when the xbox360 power supply light turns orange/yellow as it can mean that the console is off and the power supply is connected to an electricity outlet, ready to power up the console the next time you feel like gaming. If this does not happen then it could mean that the power brick is not plugged into the console properly and there could be a loose connection. Steps to take to fix this:
Ensure the console is turned off
Unplug the power brick from the wall outlet
Unplug the power brick cord from the back of the xbox360 console
Unplug the cord that runs from the power brick to the wall outlet
Wait 5-10 minutes
Plug all the plugs back together ensuring a firm connection
Plug the power brick into the wall outlet
Turn the console on and the xbox360 power supply light should change from orange to green.
Light is red - This is the worst color you want to see the xbox360 powers supply light change to. When the xbox360 power supply light is red it means that a problem has occurred and the power brick is preventing the problem detected from getting worst. You can try unplugging it and letting it cool for a few hours, and if that doesn't change the light to orange when the Xbox is off it may just be dead and need replacing.
If the light on the power supply is ok, then it may be a bigger issue with the console itself.