The fastest way to write all sectors on a drive

A few notes from my experience:

1. (the cause) If you hear an unusual sound during hd access attempts, and problems don't occur at (more or less) just random disk locations, then the root cause is most probably on the disk surface (not the electronics) - unfortunately, that's the sad scenario. If it were "just" the electronics, you might have had a chance to recover most or even all of your data.
2. (bad sectors) If you haven't already, search the web for the disk manufacturer's bootable diagnostic/recovery tool, download it, boot, run a deep test and let it try to remap bad sectors - that's the best among free methods. Note that bad sectors have a tendency to grow - so even if you manage to catch one chunk of your file after some 2314'th read attempt, chances are that those attempts just made nearby bad sectors grow, effectively decreasing the chances of recovering other parts of the file.
3. (recovering NTFS) Nothing can fix an NTFS filesystem as well as MS Windows native tools. If the NTFS image is not mountable (also make sure that you were trying to mount the partition, not the entire disk!), you can try things like testdisk under Linux, but if those fail, Windows' chkdisk can help. If you have Windows installed under a virtual machine, you can convert the raw image obtained from ddrescue to a format supported by that virtual machine (such as VDI or VMDK), add it to the VM and boot Windows in command-line mode to fix the filesystem. If you use VirtualBox, the command to convert such image is VBoxManage convertfromraw <filename> <outputfile> optionally with --format VDI|VMDK|VHD to obtain the specified output format.

sudo badblocks -v /dev/sda

but it has completed 106 hrs till now and still continue...

Hard disks do not have an unlimited life span, and it is not possible to predict exactly what it will be. If badblocks ran for that long, it is probably because of serious hardware failure. On linux, this can manifest itself at first as sudden system freezes due to processes falling into uninterruptible sleep, meaning they cannot be stopped and the kernel keeps spinning its wheels trying to handle them, maxing out the processor. This is due to disk read failures, which will be reported in syslog.

Running badblocks in that case can work but take a very long time, because it can fall pray to uninterruptible sleep too when it finds the bad blocks; if it comes out of this and logs the blocks successfully, the disk can be used, but if you reformat again that bad blocks list is gone too.

If you can't take the hard drive out yourself and examine it by plugging it into another computer, you should take it to someone who can. Taking it out is probably not that hard, which will save you a bunch of service charges including replacing the drive if necessary.