Given the windows, the view and the great floors, perhaps something like a garden style room. Plants and a seating area. Cabinets for storage along the area where you now have the children's table and chairs.

Do you need a quiet (no TV) place to curl up with a cup of tea, a good book or laptop? If so, a built-in library would be a dreamy backdrop for a couple of comfy, overstuffed chairs facing the window and a low table along with some side tables and tall lamps. A built in storage bench beneath the window with a plush cushion and pillows would give you extra seating as well as storage. If it will be a room for grownup entertaining you could have a bar on the arched wall with a mirror and candle sconces above, a curved sofa with colorful pillows, maybe even a wine refrigerator or espresso bar. I don't know your budget, but have you thought of adding a pair of French doors at the entrance to the room? They would look beautiful and further define it as its own space. Consider a larger area rug in a lighter, solid color (perhaps a natural fiber) to lift the space. Change the dark valance out for some light colored roman shades or if you don't make a window seat, frame the windows by raising the rod higher and then extending soft drapery panels from wall to wall - it will actually make the room seem larger. I couldn't tell what type of vine is currently growing outside but planting evergreens against the fence instead or maybe installing a fountain will enhance the view year round. Plant colorful annuals in tall pots that you'll be able to see through the window in the spring and summer.I know from experience how nice it feels to reclaim a grownup space for yourself - good luck and have fun!