Probably no more than 6 will be necessary (if symmetry is necessary). The lighting requirement is less than a kitchen and the fireplace side could be left unlit.

There is quite a bit of information in the LED recessed can guide for the kitchen thread.

Here is a link that might be useful: LED recessed discussion

Thanks Davidtay!

I definitely would prefer if the layout was symmetrical, but that's the concept I'm having trouble grasping with the diagonal wall (fireplace). I would also, if possible, like to incorporate a 4" gimbal spot over the fireplace to accentuate a framed picture.

I have read through a majority of your LED recessed lighting discussion, and have found it quite helpful in determining my lighting requirements for our kitchen.

A picture light might provide better and more uniform lighting for the picture than a 4" gimbal.

You could treat the space like a 13' x 11' 7" area to be lit and place the lights accordingly.
In that scenario, the nearest downlight will be ~ 5'+ away from the TV.

The ceiling fan isn't centered in the room??