The only way I can see doing what you want to do "right" is to use area lighting and an on camera flashgun. The reason I say this is that if you want to capture family images you need them to be natural, not posed--and there goes your beautiful lighting set up. The only way around moving subjects and bad lighting is to "area light" a room as evenly as possible and use your on camera flash as fill light--which is what Canon had in mind anyway. If you could truly light evenly all of your room with diffused area lighting--like having multiple softboxes in every room--you could do away with the on camera flashgun. That isn't realistic however.
My suggestion for you in your situation is to better light your rooms, maybe with a secondary light switch you can use only when you are shooting, and to use an off camera arm with the 580EX on high speed sync. Play with your shutter and stops to get the exposure settings that produce images that you find acceptable. I say this because my intense reading on the subject, and my several studio practice sessions, have taught me that really nice lighting takes lots of practice and lots of time setting up the light and making small incremental adjustments--even for the pros. This type of lighting pretty much precludes shooting small kids--they move around too much. That's why people doing family portraits with small kids--and some adults who can't sit still--use large softboxes that are really just "area lighting"--lighting the entire studio pretty much evenly. It's you typical clear, crisp mall shot--very little modeling light or other nice lighting techniques, no style, but lots of light and clear snapshot type images.
There just isn't much you can do about light when your subjects are moving around, be they in your home or out in public, except to get enough of it, or get really lucky, or both, or in the case of street people photography pick a spot where the light is where you want it, even if only for 15 minutes (rotation of the earth), and try to get a shot where the light is on the subject as you want it when the subject passes by. You can try that with your kids using your windows for key light in daytime, but put something over the window to diffuse the light--white sheet, etc. Then have your kids get near the window and try to use the light as a key. Use the 580 as fill.
Thus--more area lights, small softbox for your flashgun, play with shutter/stops/flashgun settings. Your other options is to set up a studio, but those aren't nearly as nice as on the spot snaps of a family. If you get good in a studio, you CAN bring out the personality, or as you say, the "soul," of your kids, but that is another topic. Point being that nothing can take the place of good snaps for family pictures, and your lighting options are limited.
