This is some anecdotal information without any technical references. There are ways to calculate the amount of light needed for a room, but I opted for an experimental approach and never learned how to calculate it.
I just replaced the fluorescent lights in my kitchen with recessed lights. I temporarily hung the lights from the ceiling with a cut up extension cord for power so I could test different brightnesses and color temperatures. It looked ridiculous. What I found is that I was going to need to try and match, lumen for lumen, the lights that I was removing.
There were 4 fluorescent bulbs that put out about 2000 lumens each (8000 total) positioned in an 8' strip (two sets of two) in the middle of the room. At first I assumed that I could get away with less lumens because I was spreading out the light, but what eventually happened was using 7 recessed cans (6") that each had LED drivers rated at around 1300 lumens. The cans were spaced about 5' apart. The apparent brightness of the 9100 lumen total was similar because we went with a warm white light. The harsh, cool white fluorescent lights seem a little brighter per lumen.
So how does any of this help you? Well, my advice is to experiment with lights. Hang up whatever lights you can find with a lumen rating and see what it looks like (day and night). I bought several types of lights to test and returned the ones I didn't use. Even if you just get 3-4 LED shop lights and hang them up you will get an idea of how many lumens you need. The can lights will be more spread out, but once you figure out the overall quantity of light you want, you will be in a much better place to finalize the design.
Other thoughts: Lower ceilings call for wider angle lights, so try to find the angle on any bulbs/LED modules you find. Lower ceilings can also call for tighter spacing on the lights. Spacing of 5' was good for my 9' ceilings, but you might need 4' spacing if you want a lot of light. I'd stick with 6" cans for such a large space. Color temperature is also important. You probably want a warm white (3000K) for a family area.
I found that LED modules in the 600-1000 lumen range were fairly easy to find. Anything over 1000 lumens is more rare, but available. For warm white, lights around 1300-1400 was the best I could find. Cree has 1600 lumen model (150w equivalent), but it only comes in a bright white. Also keep in mind that my goal was to get as much light as I could. A family room might be ok with less light (depending on how you use the room).