Calorimetry: Temperature Increase due to stirring of reactants
This is addressed experimentally, not through theory or computation. You run one experiment starting with the reactants, measuring the change in temperature over time. Then, you run a control with a similar solution that does not react (e.g. the product mixture), setting the mixer to the same speed. This will allow you to correct for any effect not due to the reaction. The stirring will be a minor contribution (depending on the tools used, not measurable at all). Other effects are large and always have to be corrected for (for example, if the reaction needs to be started with a spark and an ignition charge).


  What is the standard method for estimating/calculating this increase in temperature, given the rate at which the stirring occurs and all the other necessary information?


The work done to stir is based on the viscosity of the fluid. The faster you spin, the more work is necessary. In the classic experiment by Joule mentioned in the comments, a special stirring mechanism was used to have a large effect on the temperature. The experiment served to figure out the specific heat capacity of water (what energy is required to raise the temperature of a certain mass of water by a certain amount). You can read more about the experiment here.