Because recessed lights don't have any indirect component (reflected off the ceiling), you need many more of them than you do of other fixture types. This raises the cost of installation and energy use. I consider them a poor choice for general lighting.

If you have your heart set on them, I'd guess that you'll probably need about twice as many as you have shown here.

Must politely disagree with davidr. I had recessed lighting in my large basement rec/family room in my previous home and it did a wonderful job keeping the room well lit. I think your plan looks pretty good, but it would be enhanced to also have lamps on side tables that would give you lower light when someone is just sitting and reading, or when you have a fire going and want softer, lower light. You can achieve this with wall sconces, too, but lamps give you more flexiblity and I prefer to leave the walls for art.

The one thought I had on your layout is that it is not so good for people sitting and talking to each other as you are all awfully far away from each other! Perhaps moving the TV to the wall opposite the windows and the sofa to the TV spot and then bring the sofas inward off the walls a bit to create a cozier feel. The chairs could then also be moved around to face the TV if needed.