As you say yourself, you know the radius of curvature on the leading edge. You also probably have the leading-edge coordinate itself. You appear to have the equations for the airfoil, too ($x(t), z(t)$). That means you can also compute the tangent at the leading edge:
$X_{LE}=\begin{pmatrix}
x_{LE}\\
z_{LE}
\end{pmatrix}$
$\dot X_{LE}=\begin{pmatrix}
dx/dt\\
dz/dt
\end{pmatrix}_{LE}$
You might also get the tangent directly from the profile parameters, in the same way you got the curvature radius. Note I'm regarding the airfoil in (x,z) space, where x points downstream and z upwards, as is custom in aircraft aerodynamics and flight mechanics (y is the spanwise direction).
The center of the leading-edge circle is on a normal line to the leading-edge tangent, at the leading edge, at a distance of $r_{LE}$:
$X_{cLE} = X_{LE} +
\begin{pmatrix} dz/dt\\-dx/dt \end{pmatrix}_{LE}
{{r_{LE}}\over{||\dot X_{LE}||}}$
In the general case, you may not have $r_{LE}$ directly, but you can of course compute it from the parametric equations(or after-the-fact numerical analysis of a given shape, using finite differences). Instead of typing the equations for curvature radius out, I'll just refer you to Wikipedia for this:
https://en.wikipedia.org/wiki/Curvature#In_terms_of_a_general_parametrization
