Yuk.
There may be other alternatives, but it sounds like you are following the best path - take the disk out of the machine, install it in an external case and try to extract the data.
Windows doesn't like having two system disks attached, so it might be easier if you can attach the thing to either a linux or a Mac-based machine. Macs can read NTFS, but can't write to them - so not a problem.
I have no idea if the thing is failing because of the malware attack, or if it is a hardware thing. It sounds like hardware, but it seems like too much of a coincidence.
I have found with hardware issues that storing the disk in a freezer for a few hours before trying to hook it up to another machine can help to squeeze a bit of life out of an otherwise dead disk.
However, it does sound as though that sector is truly dead.
Formatting or partitioning is likely to destroy more data.
But you already knew that....
Good luck!