Modeling a 3D Wing -- Joukowsky Airfoils and Prandtl Lifting-Line Theory

I am trying to calculate the distribution of lift over the span of a rectangular, non-swept wing with a (constant) Joukowsky airfoil cross-section. The wing is rectangular in the sense that the chord length is constant. The hope is to combine results from Joukowsky airfoils and Lifting-Line theory. I have followed the derivations from the following sources:

http://brennen.caltech.edu/fluidbook/basicfluiddynamics/potentialflow/complexvariables/joukowskiairfoils.pdf

https://en.wikipedia.org/wiki/Lifting-line_theory

For the Joukowsky airfoil, the circulation is fully determined by the airfoil geometry and angle of attack by imposing the Kutta condition (finite fluid velocity at the trailing edge):

$$ Gamma = -4pi URsin (alpha + beta)$$

where $alpha$ is the angle of attack and $R$ and $beta$ determine airfoil geometry.

Now, in Lifting-Line theory, we require knowledge of the slope of the airfoil lift coefficient $C_{ell alpha}$, which I assume is $frac{dC_{ell}}{dalpha}$. However, we come up with a solution for both the induced angle of attack $alpha_i$ and the circulation $Gamma$ which have nothing to do with Joukowsky airfoils or the Kutta condition. It would appear that both circulation and the effective angle of attack (geometric angle of attack $-$ $alpha_i$) are specified now, and the above relation should not be true in general.

My question is: how is the circulation found from Lifting-Line theory consistent with results of Joukowksy airfoils, and the requirements of the Kutta condition? Can the two methods be used in tandem?