I think your problem is filtration. It appears your water is cloudy, which means there is either algae or small particles floating in the water. Since you say the water is not green, we have to assume it has fine particles in it.
Looking at your filter makes it clear you are a good DIYer. The problem is the size and design. A good rule of thumb for a small pond is that the filter should be 1/10 the volume of the pond, although I have used some half that size successfully. Yours needs to be at least twice as big as what you have.
If you look at the filters in our DIY forum or look at commercial container filters, you will find that the water flows up through the filter, not down. There is a reason for this. Upflow makes gravity your friend, while downflow makes it your enemy. When water is flowing up through a filter, particles are pulled down by gravity and stopped when they hit filter medium. If you use a "swirler," the particles settle to the bottom so well that you don't need filter pads for mechanical filtration. When I dump the water from the bottom of my upflow filters, it's black with mulm, while a glass of water from the outflow at the top looks like drinking water.
When water is going down, gravity pulls particles down. Large debris gets stopped by filter medium until it decomposes and washes down, but the tiny particles that make water cloudy just go right through. Since water is flowing out at the bottom, virtually no particles settle, so the fine stuff goes back to the pond.
Go to the DIY and look at the pinned filters. These are for 55 gallon barrels or larger, but are easily scaled down. I built my first in a 10 gallon flowerpot, which would work for your pond.
While you do need to remove the koi, I don't see any other problems. Your pump is plenty big enough for your pond. Plants are good, and with an upflow biofilter, you can put plants in pots of gravel at the top of the filter. They will absorb lots of plant/algae nutrients. Papyrus absorbs nutrients and grows very aggressively. It looks great too.