Can microfacet BRDF (including ggx) represent perfect mirror?
If this was correct, $\frac{D(H)G(L,V,H)}{4|N⋅V|}$ should be a dirac delta function. But I don't think so.


Actually, you're close to your answer - you just are trying to find out, what the assumption was in the very beginning - if you're talking about a specular BRDF. The term  $F\frac{DG}{4|N⋅V||N⋅L|}$ only works, if you start with the precondition, that you're dealing with a perfect mirror, otherwise this would be an integral itself. Have a look at PBR Diffuse Lighting for GGX+SmithMicrosurfaces by Earl Hammon Jr., where explains how to get to the specular BRDF. Especially have a look at slide 29:


Microfacet BRDF is a perfect mirror 


I.e., light reflects if and only if $m = H$


Mathematically, BRDF is a scaled dirac delta $\delta_m(H, m)$




So the perfect mirror is your model in the beginning$^1$ - but only for the specular BRDF! 

If you are trying to see, if including the diffuse BRDF into your BRDF is a perfect mirror, then this assumption does not hold any more and thus you don't have the dirac delta function.



$^1$ Note that he speaks about a microfacet being a perfect mirror. However, if all your microfacets are perfectly aligned, then your microgeometry "equals" your macrogeometry