strobe lighting is a very broad topic, and for the record I am not a studio photographer -- I am a student of light because I do product photography and "family lighting" is of great interest to me. depending on your budget and areas of interest there are a boatload of options, ranging from one speedlite to a whole raft of home studio lighting equipment. The point is that even for family stuff I find that even a few efforts can turn a snap shot into a photo. I'm not sure I understand your particular goals very well, but suspect you are investigating options and possibilities -- so I'm not claiming the below is necessarily the best option for you, just offering suggestions that I personally find to be good, and what came to my mind when I saw the picture you posted.
rather than write an epistle here let me offer an approach to learning about this particular area -- I highly recommend the two books by Syl Arena, which are available from Amazon (just search for Syl Arena). you really need to start somewhere, and all I can say is that Syl's books will give you a lot of practical and specific guidance. One of his books is "speedliters handbook" -- probably the best place to start, imho. BTW THere are a lot of folks out there selling books, but too many of them have a tendancy to show of their work without telling you how they did it. Not Syl he tells you how in a way that is both compelling and understandable.
syl's blog is at pixsylated.com
the other thing you can do right away is start reading at strobist.com. another excellent source of lighting info. read through the lighting 101 article and poke around there -- David Hobby is another one of the leaders in small flash and lighting techniques.
In general you'll discover the merits of moving your flash off camera, and modifying the light it produces. I don't know if the following approach is of interest to you (it requires more gear and set up time), but here is one of many many possible suggestions. Those that know this space will recognize that I'm endorsing Syl's choices of equipment:
put an Apollo Orbhttp://www.bhphotovideo.com/c/product/820996-REG/Westcott_2336_Apollo_Orb_36_91_4.html
with one speedlite inside, on this Manfroto light standhttp://www.amazon.com/gp/product/B0023RRPE2/ref=ox_sc_act_title_3?ie=UTF8&psc=1&smid=ATVPDKIKX0DER
with this adapterhttp://www.amazon.com/gp/product/B001ENW61I/ref=ox_sc_act_title_1?ie=UTF8&psc=1&smid=ATVPDKIKX0DER
using this hot shoe extension cordhttp://ocfgear.com/cords-for-canon-ettl/ettl-cord-extra-long/
You'll have a cord going from the hot shoe on the camera, running along the floor, up the light stand and into the Orb, where it will now impersonate the camera hot shoe (mount the speedlite there). in this configuration the camera thinks that the flash is mounted on-camera, and even though the flash is now "hidden" inside the apollo Orb, you can control it from camera menu. use full ETTL and HSS if needed. Yay for Canon for allowing flash control from the camera. big wahoo.
yes there are many fun and compelling options to this that involve even more toys. radio triggers are available to replace the cord for example. more speedlites in the Orb, other speedlites in the room, hanging from the ceiling or whatever. booms and arms to help position the Orb. All Fun, but more $$ and more complexity. For me, just getting the flash off camera and modified (with a softbox) will open up a great many possibilities and the best place to start.
Anyway, with the light coming in from the window, you could put the Orb at camera left or right, close to the subject (just out of the frame). that will produce nice soft light on the face, and the light from the window will strike the back of the head.
if you don't want to deal with light stands or soft boxes, off-camera details, etc. and you just want to light up the whole room, without concentrating light on one subject, you could probably do that with a single on-camera flash bounced off the ceiling. Or add 2nd speedlite (slaved) sitting somewhere else pointed at a wall or the ceiling.
I happen to like the speedlite method of lighting for family stuff especially because you can take full advantage of high speed sync and ETTL metering. The setup time is short, and the lights are small and portable. in the controlled chaos of a family living room, these things are great benefit. Yes there are a great many who turn off ETTL and go manual, and that is loads of fun too, but to start I just recommend moving the ETTL flash off camera into a box of some sort. Then you can cogitate on whether you want more speedlites or if you want strobes that have to plug in, whatever,
btw -- an inexpensive option for speedlite compatible flashes is Yonguo. they are getting good reviews now, particularly because they are very affordable. the 568EX for example, is essentially a 580EX clone that can act as a slave (but not a master), respects ETTL and HSS -- and its $170! at this price I can afford three, instead of getting one Canon. yes, the new Canon radio system is awesome, but again its expensive. If you already have one genuine article 580EX or 600 rt, then you just add the Yonguos to the mix. thats what I'm fixing to do...