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Maybe a low shelf/storage in front of window all the way from wall to door?
Plant, TV, lamp on it. Kids stuff inside, then you have more space on the table side?
Hang lamp over table lower with a cozy bulb.
Install indirect floodlight for lowlight craft sessions.
Agree with the votes for round table or rug.
Can what's in the hanging shelve go somewhere else?
Then you could have pictures there.
Outrageous suggestion! Pianos are so valuable for kids.
@Sheppz - personally I don't like/get the Serif. Looks like a kids chalkboard or an office flipchart/whiteboard, fussy skinny little legs like it'll fall over and it just looks odd. I think TVs look better on something more substantial, but then I don't have a TV at all so maybe not the best judge...
Think @Tenderloin 's idea of painting is a good shout. Pull a wide band of colour up the wall and across the ceiling - so either the wall on left and over the dining table (basically size of table plus seating, although pulling it wider to inc the piano is a possibility), big textile wall hanging instead of giant mirror, and timber tabletop, and you have a much cosier dining space. Or do the opposite wall and bring it across ceiling to the extent of the rug.
Of the two 'zones' which do you want to be more cosy?