One of the great challenges of fluid mechanics is that such a problem is only solvable using purely theoretical calculations in extremely limited circumstances, and the accuracy of those calculations is impossible to predict without an experimental sense of the onset of turbulence/flow separation. However, with the aid of experiments, an equation for the flow reconnection distance can be constructed with dimensional analysis.
Consider the parameters in your problem as the following:
1. Fluid density $\rho$, with units of $\frac{kg}{m^3}$.
2. Ambient flow speed in the horizontal direction $U$, with units of $\frac{m}{s}$.
3. Ambient flow pressure $p$, with units of $\frac{kg}{m\ s^2}$.
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Ambient fluid viscosity $\mu$, with units of $\frac{kg}{m\ s}$.
5. Streamwise cone length $L$, with units of $m$.
6. Transverse cone width $w$, with units of $m$.
Suppose, also, that some equation of state existed (which you may not know) such that the ambient temperature $T$ is some function of the ambient pressure and density; $T = f(p,\rho)$. In that sense, the $6$ parameters above are the only ones that can be changed in your system. If, given any such parameters, there is a unique flow reconnection distance, it must be a function of these $6$ parameters.
Because the output of the goal function $d = g(\rho,p,\mu,U,L,w)$, the flow reconnection distance, has units of length, we can describe it as depending on the following four independent combinations of the system parameters, all with length units: $$\frac{\mu}{\rho U},\ \frac{\mu U}{p},\ L,\ w$$ (These are not unique—you could come up with many functions of them that also have length units—but they are simple and convenient.)
After this, you could try to perform experiments where you independently vary each of these derived parameters to find an answer, and construct a curve to determine a numerical approximation of $d = g(\frac{\mu}{\rho U},\frac{\mu U}{p},L,w)$.
There are also some physical arguments you can make to narrow down the parameters in your system; for example, if the flow is incompressible, your function should be of the form $d = g(\frac{\mu}{\rho U},L,w)$, and if you operate in either very high or low flow speed regimes, you can even (shakily) argue that $d = g(L,w)$ only; an example of these arguments applied to pressure losses in pipes and lift/drag forces can be found in Chapters 10 and 13 of my text. 388