MacDee
I am thinking the adapter you may have is actually HDMI "over" ethernet. Just another way to transmit HDMI without using a long HDMI cable. Generally need to have a converter on the receiving device side to again convert the HDMI, depending on the adapter. (basically is a way to gain multifunction out of ethernet cables).
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For the most part, difficult to find ethernet-only routers (they do exist), so likely your router is also wifi (wireless) capable. If you login to your router, you should be able to change the SSID credentials (network name and network password) to those of daughter's last network that the RokuTV was connected to and it should connect. (as indicated by atc98092 ). Then you should be able to use the mobile app as a remote as well once your TV connects to the network.
It may be easier to just run to Walmart, Target, Best Buy and either get a Roku Voice Remote ($20) or an IR-based Universal Streaming Remote compatible with RokuTV. ($10-15).
Another method has been posted by makaiguy a few times in forum regarding using 2 mobile devices. (one is used as a hotspot with the credentials of the hotspot changed to match the previous SSID, and the other mobile device is used with the Roku mobile app connected to the same hotspot and then you can "steer" the RokuTV over to the new router network (SSID).
Refer to this thread for makaiguy detailed steps... https://community.roku.com/t5/Wi-Fi-connectivity/How-connect-to-router-after-losing-original-remote/m-p/883434
For a visual explanation of the above, you can check out this older youtube video. https://www.youtube.com/watch?v=jyhRNolATzo