Steven's sister has a 2011 MacBook Pro with a failing hard drive. She hasn't backed it up either, and there's a lot of pictures that they want to save before the hard drive goes belly up. Leo says it's good news that the computer can still see the drive and it can be mounted. The drive may be a little "messed up" and it can't read or record the data reliably. Unfortunately, in the Mac world, there aren't many good disc utilities. Disc Warrior and Tech Tool Pro have been around for quite awhile. And the more people mess with it, the less likely it will be to recover.
Leo recommends trying to image the drive and then work off the image. Hopefully, they can get all the data off that way. Another option is to take the drive out of the Mac altogether, and then connect it up to a Windows PC, where they might have better luck. Recuva could also work.
They can plug it into the PC directly using a NewerTech Adapter and then using SpinRite to fix the drive. It's not cheap at $90, but it will repair any data on damaged sectors, and move them to healthy ones. Leo also recommends changing to an SSD.