Xbox 360 controllers don't "pair" like Bluetooth devices. They use a proprietary 2.4 GHz protocol bound to the console's wireless board at factory. If it's not connecting, something is broken—not "un-synced."
Plug a wired 360 controller (or any USB HID gamepad). If it works, wireless board is suspect. If not, motherboard/USB fuse issue.
There is no "sync button" sequence. If batteries + clean board + hard reset fail, it's hardware.