I'm fairly confident this laptop, if identified correctly, doesn't have an M.2 slot since it came out more than a year before the M.2 specification was even finalized, let alone released.
I think you need to go back to the drawing board here. The fact that you're planning to take apart your laptop to run an external GPU but don't have any idea what connections you have available means you're really not ready for something like this.
Generally speaking, cannibalizing your wireless card's little PCIE slot for a GPU leads to a very janky, unpleasant experience. If you don't have a Thunderbolt slot or M.2 slot, you'll just end up turning your very old laptop into a very old desktop that will just barely run games of 2011. Most external GPU docks aren't even an option without these slots.
If you want to game and you don't already have hardware that can do that, I think you need to ditch this plan and start over with a practical goal in mind.