Here is the space from all four points of the room.

I think part of the issue is that your rug is much to small. a bigger rug will help define this seating area as a seperate room and help visually ground the couch. in some rooms you can get away with a rug that sits just under the furniture's front legs. in your space I'd prefer a rug that goes completely under all the seating area furniture. id also consider a console table to further visually mark it as a seperate space

Layout looks good, I think it is just the starkness of the rug next to the dark floors. Perhaps something with pattern that incorporates some of the beautiful sofa color. Its also too small and that is compounded by the bright color. Maybe that one could go under your dining table??

Pretty couch! I agree, you need a bigger rug.

The problem with a bigger rug is that it would extend to the media cabinet...which seems like it would look awkward? We also need to get an end table on the side by the kitchen, just need to decide on a style! What about a round jute rug?

The sofa can sit fully on the rug. If there is a little more room needed, can you move the sofa back just a bit farther? I don't think a round rug will solve the problem, or look good with the furniture arrangement. Can you use some painter's tape to mark off a larger size? What size is your current rug?

The bigger rug would go under the sofa more and a little bit towards the media cabinet but it wouldn't look awkward at all. Agree with Olychick, tape it out and see first.

If you keep the colors similar, it's not hard to mix patterns without them clashing....or you could pick a rug in a color from the ottoman that complements the sofa color. Could you post a closer up pic of the ottoman fabric so we can see the colors better? Do any of the rugs I posted above appeal to you style-wise?

Try taping out an 8x10

This is a close up of a tweedy type rug that might work for you - I just think a darker rug will cozy up the area and complement the ottoman so much better....

This rug has been discontinued and the remaining sizes are probably too big for your space, but it might be a direction to be looking, if it appeals to you:

Something like this might work, too. I can't tell for sure about the colors but Pottery Barn has samples available in store if there is one near you, refundable deposit when returned: