What is the constant in this formula for additional lift? Let’s consider the case of a plane flying s/l in a headwind with a positive gradient. A wing produces lift by inducing a bound vortex superimposed on the relative wind., and in the particular case of flying in a positive gradient, one can imagine an extra pseudo-vortex associated with the gradient itself, that re-inforces the bound vortex producing additional lift.

The question is now how to quantify that additional lift. From the variables that enter in the circulation theory of lift, one can guess that the variables involved are the density of the air rho, the airspeed v, the wing area A, the chord c and of course the wind gradient ß.

From dimensional analysis, it results that the extra lift L is:

L = k · rho · v · A · c · ß

where k is a constant to be determined. It can be probably done with a ‘material experiment’, but my question here is whether it can be derived from already known data or from a ‘mental experiment’.
The formula you have given looks strange to me, I have never seen it before. But possibly I can help you with the following brief explanation about kinematic relationships when flying under wind conditions.

Flight path velocity (VK) is the vector sum of aerodynamic velocity (VA) plus wind velocity (VW), see sketch. - When flying in a headwind condition aerodynamic velocity will increase and a pilot will need to decrease Alpha to keep the lift constant. It is the opposite way around in a tail wind condition. Now a pilot will have to increase Alpha in order to keep the lift constant and additionally will have to pay attention not to come too close to Alpha max. Or increase thrust.

If one uses the sketched kinematic relationships properly one does not need an additional formula for analysis.