How to find an unknown concentration of an a base?

when the Ka = 1.8 e-5
the pKa = 4.745

find final cnocentration
The pH of the resulting mixture was 10 units higher :  14.745
the pOH would be -0.745
[OH-] would be 10^-0.745 = 0.18 Molar

find final moles of OH
that 0.18 Molar is for a total mix of 400 ml
0.400 Litres @ 0.18 mol / litre = 0.072 moles of OH

find the moles of OH destroyed by the acid
0.200 litres of acid at 1.00 M = 0.200 moles of acid
by the equation
1 HC3H3O2  &  1 KOH  1 K C2H3O2  &  1 H2O
0.200 moles of acid destroyed 0.200 moles of KOH

so
 0.200 moles of KOH + 0.072 moles of OH remailing = 0.272 moles OH- total
the original 0.272 moles of KOH was in 200 ml
so
0.272 moles / 0.200 L = 1.36 Molar KOH

that's likely the answer they are looking for...
however the Ka has only 2 sig figs
so your answer should really be rounded off to
1.4 Molar KOH