Best DSLR + lens combo for low light, indoor kid shots

Maybe you could tell me what my price range needs to be to get a good keeper rate without too many sacrifices and compromises. If 90% of kids photos can be gotten with a budget under $1K, then my budget is probably $1K. If I have too many blurred photos and out of focus shots because the AF was slow or didn't track as well as it would have on a really high end DSLR, then my budget might need to be $2K or $3K. I don't want to spend that kind of money unless necessary but I think I should first let the task determine my budget and not the other way around.

I guess most DSLR or compacts can handle easy outdoor portraits. So I guess I want to get more info about cameras in the marginal situations. Low light. Unpredictable motion. Low light unpredictable motion. If I spend $500 or $5000 more, am I able to capture a class of photos that I wasn't able to before?

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Ted
http://www.flickriver.com/photos/tedchang/
http://www.fluidr.com/photos/tedchang
http://www.flickr.com/photos/tedchang/