I say, use Mint (Mate version), only because I personally find it a bit slicker to open the non-booting MS drive in a tab, with the backup drive as the tab to the right. I've always recovered files left to right...just seems the proper direction, somehow. Oh, and Mint looks close enough to Windows you shouldn't have an issue with it's layout.

You can Google how to download & create a bootable USB drive. Plenty of Youtubes on it. If creating from within Windows, I find Unetbootin (https://unetbootin.github.io/) does a great job of it.

Get into the habit of recovering only a few items/folders at a time, as the MS drive may be failing, it can cause the entire transfer to die & you won't know where you are then in the transfer process. At that point, you delete the partially-moved ones & start again-- with less this time. Slow & steady wins the race, as they say. Now, go & "save that data!"

Once your data is safe, you can move along to whether it's worth it to you to replace or upgrade/reinstall. I will say that I've had more than a few clients upgrade (cheap, they said) to a really anemic-by-comparison-to-old-laptop and end up having me redo that one anyway. Food for thought.