In general, the worst place to place a key light is on the camera. Why? You get a very flat and often harsh picture. It’s best to use such a light as a tiny bit of fill. Solution? Try to place your kids where there is natural light coming in windows or from room lights.
You are correct to worry about color temperature. Many posts on that topic. Do a search.
As for which light? Also many posts. Another search on the database. Or, go to bhphotovideo.com, search for “on-camera light” and sort the response from least expensive to most. Then work your way up to your comfort cost, and get that one. Some of the new LED units can be switched from tungsten to daylight quite easily. Otherwise, get a couple of sheets of 1/2 CTO or CTB depending on the native color of the light. (IF daylight, have the 1/2 CTO handy when shooting tungsten (incandescent) lights. Or, if the native color of the light is tungsten, have the 1/2 CTB to add when shooting in daylight. You also have to consider greens, which come from ordinary fluorescent lights. Get a couple of sheets of 1/4 minus green (magenta) to take out green spikes if they occur. One or two layers may be needed.
Finally, if the light does not have some kind of softening gizmo, get a couple of sheets of diffusion (I’d have some Opal when I need more intensity at the sacrifice of softness, and 1/2 grid cloth when I can go for more diffusion and less light intensity.)
Rick Wise
director of photography
San Francisco Bay Area
and part-time instructor lighting and camera
grad school, SF Academy of Art University/Film and Video
https://www.RickWiseDP.com
https://www.linkedin.com/in/rwise
email: Rick@RickWiseDP.com