Wire laptop power switch to another switch
You really need at least a basic meter to confirm which pins are which. There are four pins coming out of the switch, two on each side.  This is a SMD (Surface Mount Device) and the same pins that hold the switch on to the board are also the electrical connections.  There are several possible different configurations that the switch could be. With even a $5 budget DMM you could probe various combinations of pins to confirm which ones connect together when you activate the switch.  Then you can solder wires from your added witch across the existing switch.  

There is some likelyhood that the two pins on the button side of the switch are connected together as are the two on the rear. And then when you activate the switch the two pins on the front connect to the two pins on the back.  For example see the data sheet:

https://www.e-switch.com/product-catalog/tact/product-lines/tl1014-series-micro-miniature-right-angle-smt-tact#.VvnHluIrJEY

https://www.e-switch.com/system/asset/product_line/data_sheet/141/TL1014.pdf