Fill the plane with pentagons as tightly as possible in a regular way
If I understand the general idea correctly, you are going to close the gaps which are appearing after further structure growth. I cannot do it algorithmically but geometrically it is quite easy to do. I use Illustrator here and manually cover the gaps with two additional shapes.  

So in addition to the pentagon and flat rhombus, I will need a 5-fold star and another rhombus:  



So after first steps we get such a structure and the filling will be:  



(Initial center is marked red)  

I don't fill all holes and I leave out the obvious ones just to make arrangements more visible.

Then next step will give this:  



Here I do only halves of crack fillings, those are symmetrical. As seen those cracks indeed can be covered using same shapes without gaps and have self-similar patterns. How this can be achieved algorithmically, I don't know unfortunatelly. But this is interesting question, probably there are solutions somewhere, IDK.  

Theoretically if one analyse several crack fillings for each step, one can guess a relatively
simple algorithm.