Just saw this post. I have a a couple of ponds up and running. One has 14 koi ranging from about 10" up to about 26". The other is my Q-tank which I just finished so my 2 new koi can be quaratined for a few months. Working on another pond in my non-existant spare time.
You have a 5k pond, plumbed as a pool which means one circuit from the skimmer to the sand filter. With a electrical hog pump due to having a pressurized filter. If you have the time and inclination, I would run two separate circuits. One from the skimmer and one from a retro-fit bottom drain. (assuming your pond is concrete with no way to put in a regular bottom drain).
I would ditch the sand filter. If it's being turned off daily - you are just running your water through a cess-pool. As said above - the beneficial bacteria are just dying every time the water is cut off. Plus you are paying a lot of $$$$ for a high-head pump when you could get a smaller pump that would pay for itself probably in a year or so. Also - sand filters are good for pools, but for ponds - all they do is trap Crud which then becomes a nasty stinky mess - don't believe me - just open it up and stick your head in and sniff.
I don't know how much space you have, but 55gallon barrels make very good filters - just get ones that had food products - (I picked up several for about $15 apiece around here) - you don't want residual chemicals that leached into the barrels killing your fish.
I would run one pump from a bottom drain to barrel with some sort of media designed to mechanically filter out solids, then to another barrel with more media that is designed to be a bio-reactor. Preferably with some sort of air added as fish can never get enough oxygen in the water. Then back to the pond. If at all possible, the pump needs to be after the last barrel. The filter barrels would have to have the same water level as the pond to do this so that they would be filled by water gravity flowing into them as it is pumped out of the last filter back to the pond. Reason for this is that if pump is before the filters, then all the crud, etc is pureed by the pump impeller and is that much harder to separate out. As far as filter media - people have used all kinds, from plastic strapping, to matting, to pvc shavings. Some are cheaper than others - but some are a lot easier to clean than others and the easier it is to clean, the more it will be done.
I believe you said you had a 3/4 hp Wave II. With the setup I described above, you could switch it out for a William Lim 1/4 hp Wave I. I have one that uses something like 236 watts and pumps well over 3000 gph. I would go with 2" pvc plumbing after the pump and if you can gravity flow the filters before the pump, go with 4" pvc so you don't starve the pump.
Second circuit would be from the skimmer as now. I would ditch the high pressure, hard to clean sand filter and do two more 55 gallon barrels. Sequence is another brand of pump with a very good reputation. You can get a 1/6 or 1/8 hp model that puts out a ton of water and only draws about 150 watts. I have one on my qtank that has been working great for 3+ years.
This is only a very, brief down-n-dirty suggestion. I belong to a couple of other forums that are devoted to koi keeping. Koi are very dirty fish and can grow very large. So a good filter system is important. I also have some comet goldfish and they are nowhere near as messy as the koi.
If you are interested, I can pm a link to a koi forum. Some I would suggest, others not so much. I'm just not sure about the rules on posting a link from here.
One other thing to remember - just as I learned the term "pool-stored" after coming here, the same principle applies to ponding. They will take something, slap the word "koi" in front of the name and instantly the price trebles. And there is a world of not-so-good information out there to wade through until you can find the people that are truly knowledgable and not just wanting to sell you something.