Built-in HDD not showing on setup assist

My Macbook pro mid 2012 stopped working. I get the blinking folder with a questionmark on it. I forgot to back-up some rather important files so just reinstalling os x lion is not an option. I followed this site "If a flashing question mark appears when you start your Mac - Apple Support" to try to extract my files. Under "If Disk Utility cant repair your start-up disk" step 3, my built-in HDD is not showing. If anyone know why or an alternate way of extracting the files it would be greatly appreciated. I am able to start my mac, using the external drive, but migration assist had the same problem with finding the HDD.

I also considered just removing the HDD and connecting it to another mac, but i am unsure which adapter i should use. If anyone knows that would be greatly appreciated as well.

I have a standard 500GB drive, My mac is from mid 2012, standard specs, but with later addition of 4GB of RAM. Problem occurred shortly after upgrading to El Capitan.