No matter what you do, it is a good idea to make a backup of the system before you proceed in case something goes wrong and you need to recover.
Clean install option:
Back up your wife's personal files to a flash drive - make sure to get all of them. Then, prepare your install media in one of two ways:
1) Since it has an activated Win10 OS, you can do a clean install to the SSD using the
Microsoft media creation tool. Advantage of going with Microsoft is that it will be at version 1909 (i.e. minimal updating) and further that it won't have any Dell OEM garbage stuff installed on it. It will install most needed drivers, and what it doesn't pick up you can get from the Dell support website, OR
2) Use Dell's recovery OS download feature. The advantage of Dell's system (if you call it that) is that it will be just like it came out of the box from Dell: complete with that same old version of unpatched Win10 (meaning you have to download any feature updates and all patches that were released since then to bring it up to date, which will not be fun).
Shut down, install the SSD into the system. For now, just unplug the hard drive cables and plug them in to the SSD. Load your install media, and reboot the machine to install Win10. If you use a flash drive media, it should install pretty fast. If you don't want to associate the Win10 install with a Microsoft account, make sure your internet connection is disabled during the install.
Note:
If you decide to clean install to the SSD rather than clone the hard drive, always make sure that only the SSD is plugged in at the time you install. It is never a good idea to install Windows to a system with multiple drives as it tends to spread the Windows boot records all over the place which can cause it to fail to boot if you subsequently remove a drive.
Once the install is complete, install
Crucial's Storage Executive software which manages things like SSD firmware updates and allows you to check SSD health.
You can then reformat the old hard drive and use it for data storage. You can then copy her files from the flash drive back to it for easy access.
Clone option:
If you decide to simply clone her current installation, install
Crucial's provided Acronis software . You will need to shut down and the SSD into the system (you will need a SATA cable to do so). When you boot back up, run the Acronis software and use the cloning feature. It will copy the entire contents of her hard drive to the SSD and all files would be in their current locations. For testing purposes, once the clone operation is done shut down and simply unplug the cables from the old hard drive and plug them into the SSD. Leave the old hard drive mounted but unplugged until you are sure everything is working properly. Install
Crucial's Storage Executive software which manages things like SSD firmware updates and allows you to check SSD health. Once you are sure everything is working, shut down and re-install the old hard drive using the power and SATA cables previously used for the SSD.
You can then re-format the old hard drive and re-purpose it for auxiliary data storage for things like her pictures and work files (which you can manually move to the hard drive) from the SSD back to the empty hard drive.
To answer your prior question, the only way to get rid of the OS on the old hard drive is to reformat it (which would delete the entire contents of the disk) or manually delete the OS files (which will take like, forever and you'll have to deal with all sorts of permissions issues - reformatting is better).