How to figure out if the reaction proceeds via SN1 or SN2 mechanism? I have been assigned with the task of figuring out whether a reaction is $\ce{S_{N}1}$ or $\ce{S_{N}2}$. The information that I have is, that the yield is dependent on solvent polarity. More specifically, the higher the solvent polarity, the higher the yield. 
The problem is that both $\ce{S_{N}1}$ or $\ce{S_{N}2}$ products are the same. Both result in charge separation (therefore both are favored by polar solvents).

I proposed to repeat the experiment. I proposed to make the site of substitution chiral. That way it is possible to check for $\ce{S_{N}2}$ inversion.
Another proposal was to use the fact that the rates of $\ce{S_{N}2}$ and $\ce{S_{N}1}$ were different. However, I am not sure how to use this fact and perform an experiment.

What should I do in this situation? The people that I am working for do not want to repeat the experiment at all. Using the rates approach seems to be the best option since there won't be a need to obtain new materials. But I have no idea on how to do this. What are your suggestions? How can I figure out whether the reaction proceeded through $\ce{S_{N}1}$ or $\ce{S_{N}2}$ by using rates approach?
As you have the information that the yield depends on solvent polarity. It is the key point to be considered in analysis of type of reaction. In case of a $\ce{S_{N}1}$ reaction, the polar solvent favors the reaction. $\ce{S_{N}1}$ is a two stepped reaction, in which the 1st step is carbocation formation, such that there appears the dipole dipole interaction between the carbocation and polar solvent which favors the reaction.

Whereas in the case of $\ce{S_{N}2}$ reaction, the dipole dipole interaction develops between polar solvent and the nucleophile rather than with intermediate product. It is because the charge is dispersed in two centres in intermediate while charge is centered in single center in nucleophile. This deactivates the $\ce{S_{N}2}$ reaction and the reaction cannot proceed forward smoothly.

Since you have another specific information that the higher the solvent polarity, the higher the yield. Since higher solvent polarity favors the $\ce{S_{N}1}$ reaction, the higher will be the yield through a $\ce{S_{N}1}$ mechanism. So the reaction with tendency with higher solvent polarity and higher yield is a $\ce{S_{N}1}$ reaction.