If you already have an OS drive in the case you could toss it temporarily unformatted and try to use recovery option while already connected to the onboard sata controllers internally without the go between which may or may not still be good. That would show if the drive itself however hasn't seen an ut oh moment and gave out on you?

I doubt a sudden power outage would instantly trash a cable while that can't be ruled out for the converter you have there only the disruption despite the sata connection at one end was still usb at the other where neither should have been harmed. A glitch now being seen in Windows? Far more likely!

Now if I take the update as upgrade and the drive isn't showing up in the File Explorer(previous Windows Explorer) windows at all that indicates the drive still hasn't been initialized at all as of yet. And like I was first mentioning a trip into the Disk Management would likely be needed to get this resolved. Installing the drive temporarily in a second case would simply provide the means of directly backing things up from the drive while plugged directly into the board rather then externally by way of the usb bus since your converter lacks eSata. That provide direct access by the onboard sata controllers while a drive remains outside of the case or laptop when that option is seen.

For the fast and easy backup sticking it inside the other case would see a much faster backup from the drive. I run two external drives and the transfer rates for any volume of data fall noticeably! Even with a 128gb flash drive trying to transfer files and folders over from one desktop to another or onto a laptop at another location takes a bit of time!

But you were also indicating one other thing about that being an OS drive for the second system where simply sticking it back in where you got it from and booting from 10 media live would allow you now to finish the upgrade or even seen a clean install of 10 go on since the upgrade first is no longer required to see a 10 install activated. Clean Install Windows 10 Directly without having to Upgrade First - Windows 10 Forums

If you are not worried about softwares just personal files those should found in the Windows.old folder as far as the Program Files, Program Files(x86), and users folders are concerned. A clean install doesn't effect anything else on the drive except the ProgramData and Windows folders as well as writing over an existing Windows.old folder. Files and folders outside of those still on the drive are still found intact following any clean install since those would be non MS created folders either created by you or some app that installs outside of one of the two Program Files folders.

Even if you are only able to get a fast temp install of 10 to go on not planning to carry all the programs over you would then see immediate access to the contents on the drive other then anything system protected. You could then back things up and even consider a total wipe if found necessary to see another fresh copy of 10 go on more permanent. That would of course replace any and all of the incomplete folders from the previous upgrade attempt as well as see a fresh install of 10.