For anyone else that finds this, I ended up taking apart my computer and figuring out that the I/O chip that was connected to the power button was the thing that was making a high pitched noise. I bought a new chip from a refurbished website and replaced it, and now my computer turns on and works fine. $30 fix.  Would not recommend tinkering with your computer unless you are sure there isn't another option. For me, it was either mess around with it or throw it away, so the stakes were low. But it paid off! 
My laptop has had a host of other problems like not turning on and not starting when I need it to, but eventually it has always turned on and only made loud fan noises. But the other day when I plugged it in it started making a high pitched whine that is coming from the underside of the computer. It immediately stops when I unplug it.  Pushing the power button and holding it for any length of time has not helped. Trying to figure out if I may need to buy a new part or give up on the computer altogether.  [URL] -- This is a video I took of the sound (for some reason the sound in the video is a lot more clear when I watch on my phone)   
Dell Inspiron won't turn on, high pitched noise when plugged in