Ice, that is really a water garden type filter and not well suited for anything much past a collection of tosai. I tell you this not to insult your filter! But to say that things like cloudy water and green water are common outcomes, when you try and use a smaller filter like that with adult koi. I know the manufacturer says it is rated for up to 10,000 but that is a ridiculous statement! As a physical space, Ten thousand gallons could normally house 12 -18 adult koi . But with each one of those fish being twice as large as the bioilter chambers themselves, the possible becomes impossible ! 10,000 gallons, on the other hand , of water garden with 20-40 or so fancy comets, shubunkins ect would be a great environment with the aid of this filter.
The best media for you would be Japanese mat or matala and you could use ceramic rings or 'noodles' in one section. This would give massive surface and could cope with the water flow you are going to need.
Here is the problem with this filter besides the size- you need to have a certain amount of water passing thru this every hour to get good circulation and biological cleaning of the water. But because the surfaces are relatively small, they become mechanical in function very quickly as koi are quite dirty and produce secondary bacteria and algae growth! And as the media clog, and water flow slows, you are forced to rinse them. Brushes first and media sheets second. This can knock down and disrupt a biofilter for a bit. If you are forced to do this weekly or even more often, the biofilter bacteria has a difficult time settling in. As a result you see bacteria blooms in the water column in the form of cloudy water. Most beginners thing this is a mechanical issue or the water has 'dust in it' and try even harder to strain out the 'dust' or they try and clean the filters even better! This is not the approach as the 'dust' is bacteria making clouds as it reporduces out of control in the pond itself. In this case, the best thing to do is to stop feeding for three or four days, do NOT clean the filter and just let the biosection of the filter mature. Do very small , but frequent water changes with tap water ( treated beforehand with chlorine remover) . Trust me, one day you will go out and the water will have become clear.
For the long term, think about adding a second filter system, maybe a clarity unit like the one in the thread above yours, as a secondary biofilter and foam fractionator.
I hope you have a bottom drain? If not, do rethink that. Best of luck, JR