It seems to me that, if there are really no inert gases present, then either the 80 C must be in error or the 1 bar must be in error, or the system is not at equilibrium or something. If you really only have a binary system of KOH and water, then a pressure of 1 bar seems too high, or a temperature of 80 C seems too low.
The beginning point of calculating the amount of KOH in the vapor will be the vapor pressure of KOH. Using DIPPR's equation for liquid KOH (melting point 679 K). I get values around 10^-16 mmHg. Assuming nothing else, this means the maximum vapor concentration I would expect for KOH would be 10-16/740 or 10^0-16/200 (depending on which total pressure we use, Briezh's link suggest a vapor pressure of 200 mmHg at 175 F/80 C) which is a concentration on the order of 10^-18 mole fraction. Adding in that the KOH is diluted in water and the overall activity coefficient of KOH should be less than 1, and the actual concentration should be less. Of course, who knows how much error there is in the long extrapolation of the vapor pressure, and I really don't know what the activity coefficient of KOH is, how high do we think the concentration of KOH could be? 10^-10 mole fraction? 10^-5 mole fraction?
For the most accurate number, we would need to choose a reasonable VLE/thermo model, put together the needed inputs and parameters, then run the calculation. I don't know what VLE model you like for this. Have you thought about your desired VLE model?
Are we certain that the scrubber is there to catch KOH in the vapor phase? Could it be there to capture some form of entrained liquid (misting, atomization, splashing, plant disruption, other method that allows the liquid phase to become entrained in the vapor phase, but not actual vapor)? KOH losses due to entrained liquid (if such is possible) would almost certainly be more significant than losses due to vaporization of the KOH. Did the designer even consider the actual concentration of KOH in the vapor when installing the scrubber? It could be that the designer, knowing that the KOH vapor concentration could not be exactly 0, simply put the scrubber in there to catch the tiny amount of KOH in the vapor without really thinking about just how much KOH would be present in the vapor stream?
A lot of talk that didn't say anything, but maybe something there will suggest a next step for us.