Laptop won't charge? Or turn on at all?

I have a Toshiba C55-A laptop. I was playing Minecraft and it randomly shut off and will not come back on. It wasn't hot, so I don't think It overheated. After letting it sit, I saw the charging light wasn't on. I tried every thing I could with it. Switching cables, outlets, etc. Still nothing. Then I noticed that the cord wiggles a little in the laptop's charging port. I'm guessing I'm screwed?

Let it cool down the air vents on the laptop might have had a disorder and wait a bit or the heat ruined something inside

Well you're guessing right unfortunately. There's a possible upside, a local shop to me usually deals with just corporate accounts. I had the same thing happen and took it in there and since they have a laptop guy who specifically just does laptop work all day every day it only cost me $50.
I had to search and most places were in the $200 - $500 range, but now I just bug that dude when I have a laptop issue.
So crack open some Yellow Pages or just go to YP.com and search for computer repair, then start calling.
What's happened is the solder holding that charging port to your motherboard is broken so you need it fixed.