Most likely not Southbridge, that really hasn't been of much use in years. From the position I'd say it's the Northbridge chipset, which controls pcie links.
It's ball soldered, you will not be replacing it, that takes specialized tooling in a machine that'll run upwards of $10k for a decent one.
Only way to fix that is to replace the main board.
Laptop not turning on could be a multitude of things, most common is the battery dropped a cell so does not supply the necessary voltage.
How you can assume that one particular chip is the culprit when there's no obvious working being done is beyond me. Sounds like you were just guessing.
If it's still under warranty, contact Asus for further answers, they'll direct you for any necessary repairs. If it's out of warranty, look for a respectable repair shop. There's not much inside a laptop, but what little there is is not exactly user repairable/replaceable other than ram or storage and possibly a gpu or cpu.