>>ivank+(OP)
It is perfectly within our reach to simulate a bright daylight sky with the sun behind clouds - around 5k-10k lux - it just requires a fairly large wattage of lighting in a number of indirect fixtures. White metal halide, LEDs, and fluorescents all have efficiencies in the 50-100lumen/watt range, that just means you need to apply around 100 watts per square meter of matte white ceiling you're trying to illuminate. I like the idea of supplementing the white lighting with colored (eg: red, red-orange, green) LEDs, which can be throttled to achieve a particular color temperature.
Ideally your ceiling would be very high for this setup to give space for the light to diffuse from hanging ceiling reflectors, which is unfortunately not the case for most homes. For realistic homes, there is a style of floor lamp termed 'torchiere' that may work, though for best results you're going to want to find one with a large, fully reflective shroud - lights this bright are pretty harsh if they're not diffused well.
Set up something like this on an automatic timer to simulate the sky, and I suspect a lot of our sleep issues would go away fairly quickly.