Building eigenfunctions from eigenkets

You cannot do that because it is not correct!

What you can do instead is expressing the operator $hat{A}$ in the complete basis of the position variable $x$: $$langle x|hat A|a_krangle= int dx^prime langle x|hat A|x^primerangle langle x^prime|a_krangle = int dx^prime A(x,x^prime) u_k(x^prime)$$ where $A(x,x^prime) := langle x|hat A|x^primerangle$.

Combining this, with your derivation gives $$int dx^prime A(x,x^prime) u_k(x^prime) = a_k u_k(x)$$ which shows that $u_k(x)$ is an eigenfunction of the functional operator $int dx^prime A(x,x^prime)$. This is really nothing but your original assumption $hat A|a_krangle=a_k|a_krangle$ expressed in the basis of the position variable $x$.

Note:

The integration over $x^prime$ may seem unfamiliar, but it is necessary for the general case. In the special case of local operators, the integration actually disappears because of an implicit delta function in $A(x,x^prime)$. Take the case of momentum operator, for example, where $$ p(x,x^prime) := langle x|hat p|x^primerangle = -ihbar delta(x-x^prime) {frac {partial }{partial x}}$$ and so $$ int dx^prime p(x,x^prime) = ihbar {frac {partial }{partial x}}$$