Unless you're certain the PCIe to NVMe adapter supports being used as a boot device, you're likely better off simply installing a 2.5" SATA SSD in place of the hard drive. While NVMe drives are faster than SATA, the desktop you have is headed into its teenage years -- the boost from a SATA drive will be appreciable, and the NVMe drive almost certainly will not run at its full speed, or rather - will wind up shifting the system bottleneck to the CPU and RAM.  
Will this adapter work in our old Inspiron 560? I'm reasonably sure it will but, not positive, looking for a more experienced take on the matter. At this point, it's mainly my wife's computer, I know she gets aggravated with how it seems to be bogging down more and more on the old HDD and whenever I get near it it seems painfully slow compared to my laptop, so I'd like to update it for her. Would one of these drives be a decent fit with that older system? Can't remember which processor it has but, I do know that it has 8GB of RAM and is maxed out there. She doesn't do anything that requires a lot of drive space or incredible speeds but, it would be nice to get things working a little better for her. Any input on the best fit for this system would be greatly appreciated.
Inspiron 560 Upgrade