Maybe I can explain it a different way, sort of it explain the basic framework of how a Sonos system works. At the top level, you’ll set a Sonos system, or household. A system contains Rooms, sometimes called zoned. A room is made up of a single Sonos speaker or device (amp or port) or multiple Sonos speakers semi-permanently wirelessly bonded together. Examples are adding a sub to a soundbar, or rear surround speakers to a soundbar, or two speakers making a stereo pair. Each room can play it’s on audio source, typically streaming services or TV audio, or be temporarily grouped together with other rooms to play the same same audio source...all wirelessly. When that audio source is a streaming service or aux input...music...the audio will play in perfect sync across all rooms. When the audio source is TV audio, the room connected to the TV plays the audio immediately, to match the TV video, while other rooms will be delayed slightly, as multiroom audio requires some buffering to work well.
To answer your question - I want the speakers in the kitchen to duplicate the sound from the TV. How do I connect the TV and the Speakers together?
An Arc would be connected to the TV as one room, with the amp being setup as a second room (kitchen). You can then group them together to the TV audio together, but as I stated before, the audio coming from the kitchen speakers will be slight behind the audio coming from the Arc. If your kitchen and living room are not in an open concept space, it would be ok as you wouldn’t hear an echo effect from the two rooms in the same space. But they are in the space, so it won’t work well.
Separate but related question: Could I leave the speaker terminals in the HT Room 2 with the Sonos Amp and still connect it to the TV in Room 1?
Yes. the amp (physically sitting in HT Room 2, but speakers in kitchen) is one Sonos room, while the Arc (or whatever you use) in the living room is a 2nd Sonos room. Therefore, you can temporarily group them them together to play the same audio source. Again, if it’s TV audio, the two rooms won’t be in sync.
There is a more complicated, wired, way you could do what want, but even that isn’t a perfect sync. That would be to get a a HDM switch between your TV sources and TV that has an optical audio output as well as HDMI output to your TV. Your TV is then connected to the Arc, and you have your living room audio covered like normal. The optical audio output then needs to be wired (via Sonos optical/HDMI-ARC adapter) to the Sonos Amp sitting in your HT room 2, so it can directly play the TV audio as well. The result is both are playing the same TV audio stream, but not at all using Sonos to group the two rooms together. The streams are likely to be a few ms off each other, and might still have an echo, not sure. However, if you’re going this route, you could do this with any soundbar or AVR really, you don’t need to use Sonos products, since you aren’t really using Sonos multiroom audio features.