How to calculate resistivity from this cell design?
Apologies in advance, because I just know that this is not the answer you were hoping for, but truly the correct answer is "you don't".

In practice people determine the cell constant via calibration. See for example:
Page 20 of Conductivity Theory and Practice
Pages 14-15 of the YSI Model 31 Conductivity Bridge Manual

Standard practice is to have a calibrated solution (to be clear, the word "solution" is here used in the sense of "a liquid containing a dissolved solute", not "mathematical answer"...) with a known conductivity then measure it with the cell. Since the cell just measures conductance, you use the (measured) conductance and the (known) conductivity to back-calculate the cell constant.

In theory you could calculate the theoretical cell constant as you suggest, but:  


You've only calculated the perpendicular faces. There are fringing fields that will affect your measurement too. Good luck calculating those. It's not impossible, but you'll need a good finite element modeling tool or something like that.
Your answer will only be as good as your measurements of the physical dimensions.
There are some effects (e.g. accumulation of fouling, plating effects) that will affect your cell constant, that will not be accounted for in your equations.


When all is said and done you can do a whole lot of work to calculate a theoretical cell constant that will be only modestly accurate for the reasons outlined above, or you can use a calibrated solution (like the scientists do) and get a much more accurate result. Granted, it's still a whole lot of work, but the end result is decidedly better.