Huygens' principle seemed quite complete in this description if applied correctly. I noticed that Fresnel was credited in the Huygens-Fresnel principle, as well. I think it is because he was able to describe diffraction with the principle correctly, something Huygens could not do in his lifetime. The correct description of diffraction using Huygens' principle took ~120 years after Huygens' death. It is quite curious to me.
Why couldn't Huygens explain diffraction using his principle?
Because he did not think of wave fronts as "fronts of amplitudes" that can go both ways, and cancel each other when superimposed. Sometimes it is hard to imagine how simple ideas could be missed in retrospect, but perhaps they are "simple" to us only because we already know to pay attention. Huygens thought from what later came to be called the optico-mechanical analogy, see Did wave optics anticipate quantum mechanics? So he paid attention to the parallels between the spreading of waves and light (or particle) rays rather than the differences between them, on reproducing the effects of geometric optics. The time of focusing on wave effects came only later, according to Mathpages:
"Subsequently, Augustin Fresnel (1788-1827) elaborated on Huygens' Principle by stating that the amplitude of the wave at any given point equals the superposition of the amplitudes of all the secondary wavelets at that point (with the understanding that the wavelets have the same frequency as the original wave)."
Are you sure he did not try? Have you read Huygens? His contemporary Newton did try to explain it in his Optics. Anyway, if Huygens never addressed this question, it is hard to tell why.
But it is easier to explain why it took 120 years before the correct explanation was given. The reason is that the wave theory was not popular until the end of 18th century. The prevalent theory was corpuscular, and it is hard to give a correct explanation of diffraction within corpuscular theory. (Newton tried). One of the reasons corpuscular theory theory was almost universally accepted for so long was the enormous authority of Newton, though reading Newton indicates that he was not so sure.
