How to repair or recover data from a dropped portable external hard drive

You could try ddrescue:

GNU ddrescue is a data recovery tool. It copies data from one file or block device (hard disc, cdrom, etc) to another, trying hard to rescue data in case of read errors.

A good primer for ddrescue is on the ForensicsWiki. Install it on Ubuntu by running sudo apt-get install gddrescue. Next, you'll recover the partition. Quoting from the article:

First you copy as much data as possible, without retrying or splitting sectors: ddrescue --no-split /dev/hda1 imagefile logfile

Now let it retry previous errors 3 times, using uncached reads: ddrescue --direct --max-retries=3 /dev/hda1 imagefile logfile

If that fails you can try again but retrimmed, so it tries to reread full sectors: ddrescue --direct --retrim --max-retries=3 /dev/hda1 imagefile logfile

The imagefile is going to be as big as the partition you want to recover, so make sure you have plenty of hard drive space (obviously not on the same HD you are recovering from).

Replace /dev/hda1 with the partition (ie /dev/sda2) you are trying to recover. Once you do this, you can mount the imagefile like this:

sudo mount -t ntfs-3g -o loop,ro imagefile /mnt

This mounts the image read-only at the directory /mnt.