First thing I would do is verify the health of this hard drive. If the drive is failing, then you need to know how badly in order to determine if there is anything you can do, or whether you should contact a professional data recovery service. Since the drive is formatted using HFS+, there is a chance using the paid app Disk Warrior could fix the issues assuming the drive is physically healthy since Disk Warrior will not work on a drive it deems is physically failing, so you would want to verify the health of the drive first before purchasing Disk Warrior.
You can check the health of the drive by running DriveDx (free trial period) and posting the complete text report here using the "Additional Text" icon which looks like a piece of paper. I'm not sure if it will be able to access the health information when this "broken" laptop put in Target Disk Mode is connected to a newer Mac. It is possible to create & boot from a Linux USB stick to check the health of the hard drive on the 2011 laptop....just let me know if you are interested in the instructions.
You can try running the Apple Diagnostics to see if any hardware issues are detected, but many times it won't report failing drives even when the drive is actually failing. Plus it won't tell you how bad the drive failure is.
If you don't have backups, then make sure you start backing up your new computer since there are a lot more new ways to permanently lose access to the data on the more recent Macs due to all the hardware, software, and security changes. Apple includes Time Machine backup software with every copy of macOS to make it easy for people to have backups....the user just needs to purchase an external drive.
