Your question has many things duplicated. If the question should read: "what volume of 0.250 M KOH is needed to completely react with 10.6 mL of 0.235M H2SO4" then here is the answer.
You will need the balanced equation which gives that 2 moles of KOH reacts with 1 mol H2SO4. Given the volume and concentration (molarity of any solution you can calculate the moles of solute (reactant, in this case H2SO4. Using the mole ratio relationship from the balanced equation, you will be able to determine the number of moles of the 2nd reactant (KOH) which will be needed to react with it.
Remembering that when we are given the molarity of a solution it is telling the number of moles or the solute (KOH) present in 1 liter (1000 mL) of the solution, the volume of the solution can then be determined as follows:
(0.0106 L )(0.235 mol H2SO4 /L soln) ( 2 moles KOH/1 mol H2SO4) = 0.004982 mol KOH
Volume of KOH = 0.004982 mol KOH ) ( 1000 mL / 0.250 mol KOH ) = 19.9 mL
Explanation given above CAN be used to help solve ANY solution stoichiometry problem, not JUST this one. Hope this helps!