Let's get real.
1. There is nothing more annoying to subjects than having a flashlight, let alone an incredibly bright LED, aimed at them. Your kids will rebel, and forget guests. It is bad enough pointing cameras at family memebrs and getting them to keep doing what they are doing. And you could blind them.
2.There is nothing worse that the light from a camera for illumination, unless you are shooting a horror video, in terms of video quality.
Forget camera lights; maybe acceptable at weddings where newlyweds, and even their guests, seem to put up with all sorts of camera incoveniences to have a record of their happy day.
I can't know how badly lit your home is. But if you cannot get decent videos in your home with $1000 dollar camcorders (which ones?), then you need to improve your home lighting, not just for photography, but for your kids' eyes (and yours).
You might consider the Panasonic LX7 for video; it takes good low-light video. Check out the thread on that. The Sony RX100 is another (much more expensive) alternative for dim light - they have a fast lens/bigger sensor than camcorders, but limited zoom. they also enable you to actually control ISO, shutter, and aperture independently (unlike most camcorders), so you deal with low light issues more intelligently.