The calorimeter constant is a conversion factor that relates the temperature change observed to the heat released in the chemical process being measured. The idea is to determine how much energy is being released per degree of temperature increase observed.
In this case, you need to determine ho much energy is released in the reaction. Once you have that value, you divide by the change in temperature to get your ratio.
Calculating the energy released is basically a limiting reagent problem. Whichever of HCl or NaOH is the limiting reagent will determine how many moles react, and just how many kJ are liberated in the reaction. So - let's have at it, shall we?
(0.050 L)(1.0 mol HCl/L) = 0.050 mol HCl
(0.050 L)(1.2 mol NaOH)/L) = 0.060 mol NaOH
Since these react in a 1:1 ratio, HCl is clearly the limiting reagent. So we can use 0.050 mol HCl to calculare the number of kJ released in the reaction.
(58.3 kJ/mol)(0.050 mol) = 2.915 kJ (good to two sig. fig.)
Now we divide this by the temperature change.
(2.915 kJ)/(25.10 C - 21.07 C) = 0.7233 kJ/C (again, good to two sig. fig.)
So this means that any process observed in the calorimeter will produce 0.72 kJ of energy for each C of temperature increase observed. Now this is only true if the calorimeter has the exact composition under which the constant was determined. But since the majority of the material in the calorimeter is water, it will give reasonable results for other reactions observed at similar final concentrations.