Finding Spectral Illuminance from total Illuminance

I'm trying to find the "Spectral" Illuminance (lux/nm) of a LED in order to find the spectral irradiance (watt/nm).

I am somewhat new to optics, especially when it comes to the radiometric and photometric units.

I understand that the Lux is just the watt/M^2 scaled to the human eye using the luminous efficacy for each wavelength.

I want to know if this method is correct.

What I have----
The relative luminosity function (call it L(λ)) lm^-1

The total Illuminance from the source (~8 lux)

What I think I need to do----

Integrate my L(λ) function multiply by C and set that equal to 8
C is this "normalization" constant

(I based this off of the fact that if I have 3 lux of light A and 4 lux of light B on a surface, I will have 7 lux total. In this case A and B are different colours, and I have a full continuous spectrum of them ~350nm-600nm)

Then the spectral Illuminance call it S(λ)
is just C* L(λ)

now I can take this function, C* L(λ) and divide it by the luminous efficacy function η(λ)

finally my spectral irradiance function E(λ) is =C* L(λ) / η(λ)

Physically this equation incorporates the LED characteristics, the measured light, and the whole lumen/watt : photo/raido metric stuff.

Could someone tell me if this is correct and whether or not I missed something?

Thanks