With an ideal light source for slide duplication in mind (better than Illumitran) I have similar questions to answer.
Any LED solution right now will have spikes in the spectral distribution. Whether based on RGB LEDS or white shifting fluorescents on UV LEDS. The continuous spectral distribution white LEDs filtered by the smallest Quantum Dots possible are not marketed yet. Their Kelvin grade can vary depending on some tweaks and voltage difference. Right now low voltage halogens from 2800K to 5000K are the only option for continuous spectral output and different color temperatures.
Building something like Philips did but better will take time. You could at least check the spectral output of the RGB LEDs and whether they fit the camera sensor sensitivity best, of both data is on the web. There will be an electronic scheme somewhere on the web too, tinkerers like to replicate what is on the market. In the disco scene there will be amateur up to pro equipment like that.
Another somewhat rougher approach is your iPADdisplay trick translated to two small secondhand flatscreens, a PC and two videocards. Color per screen and gradations per screen etc. possible. Someone must have done that before and open source software could be available. Spikes are still there from the LCDs backlight.
Met vriendelijke groet, Ernsthttp://www.pigment-print.com/spectralplots/spectrumviz_1.htm
December 2012, 500+ inkjet media white spectral plots.