Even with a very narrow table of 36", you need 24" on either side to pull out chairs. That alone is 7', which only leaves you 6' for a sofa and coffee table. Your best bet is to lay out measurements of the sofa with blue tape and see what you have left for space. Is there a reason you need to fit both areas into one space?

I would use the side of the room where you currently have your sofas as the "adult" space, and the corner where you have the play rug as the "kids" space. You need to separate the two. So move the desk to the adult area - either on wall to the right side of the opening to the family room, or slide the sofa under the window closer to the other one and put the desk on the wall between the sofa and entry. Move the two tall storage units to the staircase wall, and use them exclusively for toy and game storage. The kids kitchen can go on the wall to the left of the doorway opening. From your entry, you will see mostly the adult area, and the messier toys and all the shelves won't be immediately visible. One other option that could work depending on your floor plan is combining your living room and dining room in this space, and using the dining room exclusively as a playroom.

Here are some ideas for an elegant sitting room: This is called a reading alcove! I found this a very elegant idea due to the curved sofa, wall color, and other decor.

Two quick thoughts... One: if you would add dimensions to your sketch it would be helpful. It wasn’t apparent to me where the piano fit on your floor plan. Seeing the photo of your room is helpful, but still doesn’t give accurate information about wall space, doorways, and the size of the window. Two: I don’t play but that looks like a cramped, uncomfortable place for the piano. It seems to me to be the least likely place for listening and playing. If I understand you, you want four really comfortable chairs arranged in a circle around a beautiful cocktail table. I’m sure the pros here can suggest upholstery, a stunning rug, and a table to give you a beautiful space. Just please, please go to a furniture store and sit in any chair you are considering. I will never understand buying chairs, sofas, and mattresses online. Of course we have back issues in our family so I take a long time to purchase these items so we can sit in comfort. I also have a SIL who is a tall and muscular size so I buy dining room chairs for his comfort, not dainty darling chairs for looks. The secret to entertaining is comfort, personally, I would pick the upholstery first because the four chairs will be the big expense, then pick a wonderful rug to anchor your space. The piano needs to be moved to space where some cabinets or shelves can anchor and enhance it. My DD has a beautiful antique jelly cupboard in original blue paint by her baby grand to give height to the space and serve to store her music. In your case, it could also house cocktail napkins, trays, etc for your evening entertaining.