Xbox 360 console power doesn't work

Are both console and the Power Supply in a well ventilated area? A overheating Power Supply can sometimes cause your Xbox 360 (or Xbox 360 Slim) to freeze, make sure the console and Power Supply are in a well ventilated area, if this doesn't work or both
console and Power Supply are in a well ventilated area, try the Freezing Screen Solution in the Xbox 360 console support area, if the Freezing Screen Solution doesn't solve you problem your console may need to be repaired or replaced.

Hello dear Gamer, first it's a pleasure to be able to help you today.

My name is Vinicius.

I am a gamer and independent advisor on the XBOX community platform.

Hey Nate, how are you today? Hope you're well.

Since it's not showing anything on the screen, all the possible fixes are reduced. So, It might be something going on into the hardware of the console, since it's not syncing with the controllers, and not even changing the HDMI cable is working. I advise you to take your console to any repair store in your City to an analyzes on it.

Well, I went out and got a new HDMI cable. Problem solved. Unusual solution, but a working one. Thank you.

most likely the HDMI port or cable, I remember with he 360 if it does not detect a connected screen then it does not fully boot and therefore the controllers would not connect

Try the steps listed
here. If it doesn't work you will need to send it in for servicing. As it's only 3 months old, it shouldn't cost you anything.