The Dirac delta function can be defined as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{itx}dt \, .$$ From this we see that the dirac function has dimensions of $x^{-1}$.

How do we represent the dimensions in cases like the momentum eigenvectors which, when units are included, is represented as $$\frac{1}{\sqrt{2\pi\hbar\cdot(kg^{-1}m^{-1}s)}}e^{i px / \hbar}$$ or $$\frac{1}{\sqrt{2\pi\hbar}}e^{i px / \hbar} (kg^{\frac{1}{2}}m^{\frac{1}{2}}s^{-\frac{1}{2}}) \, ?$$

Is there a preferred way to write the units (not constrained to SI units, any other system including natural units too) or do we just leave them out, though it would be dimensionally inconsistent without implied units. Sources I can find for the momentum eigenvector ignore the units of the delta function without even mentioning.