My daughter has a TCL TS 40S325 rokuTV. Remote stopped working so she was using phone app. Then she moved new ISP router/modem only has ethernet ports. Phone app won't connect since modem hasn't assigned it an IP. TV only has HDMI ports. Tried a HDMI to Ethernet adapter still no IP showing up as being assigned to subnet. Also tried a mouse and a keyboard into TV USB to get thru TV setup no luck. TV only has one button and doesn't do much other than turn on/off TV. New remote on order. Verified no IR light coming from old remote.
If she has access to the WiFi router, she could change the SSID and password to match her previous network. If not, you have to have a remote. There's no way to use the app until the Roku is connected to the network. I've never seen or heard of an HDMI-Ethernet adapter, but I see such a thing exists. The problem is the Roku device has to have the drivers for such an adapter, and they don't so there's no way to use one.
However, the TV does support an IR remote, so you could grab an inexpensive universal IR remote that supports Roku TVs and use that to get connected. I've seen many such remotes available anywhere that sells electronic stuff like that for under $10.
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/...
For a visual explanation of the above, you can check out this older youtube video. https://www.youtube.com/watch?v=jyhRNolATzo
Yeah, after I posted my response, I noticed that the HDMI Ethernet adapters that popped up on my search were merely HDMI extenders, not adapters that can be used for an wired network connection.
There was the failed attempt at HEC technology (HDMI Ethernet Channel) that attempted to do just this. Never took off. It is kind of like Bigfoot. People swear that devices with it existed, but never actually saw any in person.
I do know that they have one form of Ethernet over HDMI. It's to permit a TV to use an Ethernet connection on an AVR. It requires a supporting Ethernet cable, and of course both the TV and AVR have to support the functionality. I don't think it ever really caught on, and I doubt the OP's TV supports it.