We are digital librarians. Among us are represented the various reasons to keep data -- legal requirements, competitive requirements, uncertainty of permanence of cloud services, distaste for transmitting your data externally (e.g. government or corporate espionage), cultural and familial archivists, internet collapse preppers, and people who do it themselves so they're sure it's done right. Everyone has their reasons for curating the data they have decided to keep (either forever or For A Damn Long Time (tm) ). Along the way we have sought out like-minded individuals to exchange strategies, war stories, and cautionary tales of failures.
Data recovery is not really something you should DIY unless you are fine with losing the data. Chances are high you will make professional recovery more expensive or outright impossible by tinkering yourself.
Your best bet is to make a disk image. For this the HDD will be read like a vinyl record from beginning to end. For a file copy the read head usually has to seek around a lot to read all the file (fragments) bit by bit.
Does it show up on the system? If you want to diy it I'd give ddrescue a shot and see if it can get most of the data. DDrescue is made for failing drives and can restart, and try to get bad parts of the drive later on.
You've changed the sata cables and tried a different port yes?