Marla,
Did you install Windows 10 directly or did you install some previous Windows version and then upgrade to Windows 10?
If you used the upgrade path (which I did), then you will need a new license. Upgrades are tied to the hardware on which they are installed. Change the hardware and you need a new license (in my case, I had to reinstall Windows, but this was because I had no additional Windows 10 licenses and had to use yet another license from a previous Windows version).
If you used a Windows 10 license, provided that you (at least once) logged into a Microsoft account using the previous KY NUC, the licensing should be ok. Simply install the SSD(s) and SODIMM(s) into the new unit and boot. Once you log into the account, if it doesn't reactivate automatically (since I didn't go through this myself, I am unsure about this part), you can simply reactivate it yourself (you aren't supposed to have to reenter the original license key, but have it handy just in case).
Hope this helps,
...S