You are lucky you have such big contact areas there. But yes that stray wire is making contact and essentially shorting out your power button. So the laptops power button is essentially now always pushed down.
If you have not done so already, take your battery out and make sure the laptop is not plugged in. Just in case there is a trickle of power that shouldnt be, and it should never be live when doing anything inside or a stray wire will be the least of your problems.
Get some desoldering braid put it ontop of the joint you made, and put the iron ontop. It will wick up the solder and desolder the joint.
Then try again. Dont leave the soldering iron there for too long because you will heat everything up, and these parts arent meant for that sort of heat for long periods. They are soldered by machines that heat them for a second and done. Not 5 minutes with a big blunt tip.
Also that wire is way too large. Trim it down or something. And get some insulation for it. Even if it did work after this run, it wouldnt have later because not all your strands are on the solder area, and are loose.
Good first try, but you should really practice on something less expensive. You can and will destroy your laptop if you are not careful.