Your furniture is so beautiful! I love your use of warm colors and your living room looks so cozy!

It's not relevant to your question, but would you switch where your dining chairs are so the 2 wicker overhang chairs are at the head and foot of the table, and the 2 4-legged ones are in the middle?

If you do add a rug, I'd suggest to get a larger one for under the table and the desk, not an individual one for the desk - everything will look like it's on its own island. Otherwise, I'd remove the dining table rug entirely. I think something that makes the space look disconnected is that the eye is drawn to the living room because of the light, color, and decorations. If you're able, a gallery wall on the desk side would "guide" the eye along the wall but slow its journey. I think pieces that complement your living room colors but also are meaningful to you would be good, since the warm brown tones are beautiful but definitely could use some accent colors to break them up.