Windows Server 2016 is not listed as a supported O/S for the KY NUC, so it is unlikely anyone has seen this issue before. My suggestion (which you will likely not like) is to install an different (supported) version of Windows and verify whether or not the LAN port is working then (Intel Customer Support (ICS) will also likely ask you to do this before authorizing a replacement). If it is not working then, contact ICS and arrange for a RMA replacement.
Hope this helps,
Server 2016 is listed as a supported operating system for the NUC6i7KYK on this page: Supported Operating Systems for Intel® NUC Products with the caveat that Bluetooth, Wireless, RAID, and Thunderbolt are not supported.
Edited to add link to Server 2016 driver: Download Intel® Gigabit Ethernet Network Connection Driver for Windows Server* for Intel® NUC Kit NUC6i7KYK
Download the network driver from Intel: Download Intel® Network Adapter Driver for Windows Server 2016*
Uninstall the current installed driver and install the Intel driver.
If the driver will fail during installation (i.e. you will see the same massage: Error "Cannot install drivers. No Intel(R) Adapters are present in this computer"), try to install the driver manually step by step. Follow these steps:
1. The driver file is in .exe format. Extract and decompress the file to a folder. (To extract the file, you need an archive utility such as 7zip. If you don’t have one, you can download it online.)
2. Go to Device Manager.
3. Expand category “Network adapters”. Right-click on the Intel network device and select Update Driver Software…
Note if the network driver is missing, it might be listed under category “Other devices”. In this case, there is a yellow mark next to the device which is displayed as Ethernet Controller.
4. Click Browser my computer for driver software.
5. Click Let me pick from a list of device drivers on my computer.
6. Check the box next to “Show compatible hardware” and click Have Disk… button.
7. Click Browse… button and navigate to the folder where you extracted the driver file to.
8. Continue to open the folders until you can choose any of the .INF files. Then follow the on-screen instructions to install the driver.
Can you please try version 22.1? Here is the url: Download Intel® Network Adapter Driver for Windows Server 2016*
Did you try Windows 10? If you are having the same issue with Windows 10 then I would say you have a defective unit.