BEHOLD:
I am talking to you now from my laptop! It works!
The problem was nothing like I thought it would be... although I didn't think much since I had absolutely no idea what it WAS. But I didn't think it was THIS:
I unscrewed every screw in my laptop, putting each one on one of several rows of tape I had carefully organized to reflect the back of my laptop, so that I knew where each screw went. I then took off every cap that was meant to detach from below it, and turned my laptop right site-up. The hard drive stumbled out, attached to my laptop by a wire. So I looked underneath that.
Inside I saw two wires, each with a rather bulky, metal head. One wire was clamped to a chip by the head. The other wire was loose. So I re-attached the wire to a part of the chip, judging by the connection pattern underneath the wire and the only part of the chip that could fit it.
And then I found I was able to turn the machine on.
In other words, it was just a loose wire that needed re-attaching.
But, this has woken me up! I need to start backing up my files, because my laptop needs a lot of work done.