I just had a quick run-through with Trinity Rescue kit. It will perhaps be too much for a non-Linux user. That is my first impression.
So then, I have come to the best way to clone a failing HDD sector by sector atleast for now

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Tool: ddrescue Source: SystemRescueCD
1. Download the SystemRescueCD from
Download - SystemRescueCd systemrescuecd-x86-4.2.0.iso 380MB and create a bootable pendrive with that ISO using Rufus
Rufus - Create bootable USB drives the easy way
2. Boot your computer with that pen drive with your source disk (your failing disk) and a destination disk (formatted empty disk of capacity not less than that of your source disk) plugged in.
3.Against the command prompt type: fdisk -l [ENTER] This will list all your drives. Note the nomenclature of your source disk and destination disk.
4. Next against the command prompt type: ddrescue -r3 -n -S -v /dev/sdb /dev/sdc recovery.log [ENTER]
Note: Replace sdb and sdc with the correct device nomenclature of Source disk and Destination disk as obtained in step 3.
5. Sit back and relax. The cloning process may take days depending upon how bad your Source disk is. If your data is important patience is the key and the wait worth it. The saving grace is you can stop the process any time with Ctrl -C and then restart. It will take the cue from the log file and start from where it left.
In the cloning process itself it will go through cloning all the good sectors first and then try to recover the bad sectors one by one trying it 3 times (r3 option) before giving up on it.
I have actually tried it cloning an 8GB pen drive with bootable Peppermint Linux ( that was created sometime back on a request from Golden to try it and report ) onto a 32GB freshly formatted pen drive. It took about half an hour. At the end of the cloning I was able to boot into Peppermint from that cloned drive which signified successful sector by sector cloning including the boot sector/s.
May be Anshad Edavana will be able to polish it and improve upon it as one who had actually tried it on a failing disk. ( As indicated already I had no failing disk.)