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So I was looking at the graph of the white LED, intensity vs wavelength and I observed two peaks. enter image description here

Which peak would I choose in the formula Eg = 1240 / lambda?

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    \$\begingroup\$ I suspect the phosphorescence has nothing to do with the internal working of the LED - it's possibly a coating that is activated by the blue light from the LED and in turn produces other colours to give an illusion of white. I could be totally wrong of course. \$\endgroup\$ – Andy aka Aug 11 '15 at 14:41
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Quote from here, a pdf article on the subject: -

enter image description here

This means you use the blue luminescence wave length for your calculation.

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