As dstarr3 suggested, getting a 2.5/3.5 SATA kit would be the smartest move. It was my mistake to give you that link, I was just trying to point out what kind of cable/adapter you're looking for.
A new case is the solution here, but if you have the tools, you could try removing the edge that makes it impossible for you to insert the graphics card.
I would do that while waiting to buy a new case.. just a temporary workaround if it's possible.
You'll need a sata-to-usb cable to be able to connect your hard drive to your laptop and access your files.
https://www.amazon.com/StarTech-SATA-Drive-Adapter-Cable/dp/B00HJZJI84
.. that is the easiest method.
You could install your hard drive to your computer, but I don't recommend that.