First things first:
If you want your data. DON'T do any writing to the card as you try the rescue.
I just happened to have the same problem on a friends PC with an external HDD which got its volume entries corrupted and was also only showing a few MB instead of 60 GB. This happened because he did not safely remove the drive from his PC. If your card is really broken then nothing except a professional rescue service will be of any help to get your data.
Step 1: open a terminal window
Simply press ctrl+alt+t to open a terminal window.
Step 2: acquire testdisk
testdisk
is available in the normal Ubuntu repositories so simply install it by:
sudo apt-get install testdisk
Step 3: rescuing your data via testdisk
Insert your SD-card into your card-reader or plug in the drive you want to rescue and fire up teskdisk with:
sudo testdisk
Then you will see the following in your terminal window:

Here you can choose to create a log file which I highly recommend for eventual troubleshooting. Simply press enter and you will see the next in your terminal, where you can choose the drive you want to rescue:

Of course you see my drives in the screen-shot so you have to navigate to the corresponding drive with the up and down arrow key and then select at the bottom proceed with the left and right arrow key and hit return.
Next you will see is the file-system selection like in the next screen-shot:

In most cases it tells you in the hint at the bottom which file-system type to select, do so or set it to the corresponding type of your drive. Then hit return again to proceed.
In the next menu (see screen-shot) select 'Advanced' and you should be able to see the full partitions (maybe a second partition on your card). Select it and select 'List' files and back them up.

You should then see the directory structure you had on your drive like in the next screen-shot:

See the red ones on the screen-shot; these are deleted files you can choose to hide by pressing h. Now navigate to the files/directories you want to rescue by pressing up and down arrow key (you can switch directories with enter or go upward by selecting the ..
entries.
To make a selection of which files you want to copy/rescue press : which will mark the corresponding row green or press c to copy a single file. For copying the selection press shift+c. In both cases you will then be prompted with another directory view where you can navigate to a place to save your data/files. Navigate to where you want to have this data and press shift+c again to start copying.
Step 4: cleaning your SD-card
After you have all your data rescued its time to clean up the SD-card so you can use it again to full capacity. To do this just enter the following in your terminal (make sure you select the right volume for your card which was sdb
in your screen-shot so I take that for my example here)
sudo dd bs=4m if=/dev/zero of=/dev/sdb
This may take a while to run but after that there is nothing on the SD-card at all so you should be able to use it with your reader and simply format it again.
sudo lsblk
andsudo fdisk -l
. – Andrea Lazzarotto Oct 31 '16 at 22:47