What is in the two big filter tubs - do they have filter media on the bottom? By filter media this could be anything from loose sponges to lava stone to specifically shaped bits of plastic. These are all there to provide surfaces for the good guys to live on and process the incoming waste. Alternatively, maybe the previous owner was relying entirely on biological filtration, ie plants with roots with nitrobacter on the surfaces. The problem is that there are many variables that can effect the overall efficiency of your cycle, all of which may have an incremental impact.
For what it's worth I have something different, a closed pressure filter with pads, and I basically just replace them every year. So pump sucks in water, gets put through a loop with filter pads, the filter pads physically trap the waste, and the nitrobacters grow there to convert it, and there's an internal UV tube which kill off algae continuously, water comes out the other end cleaner over waterfall, and cycle goes on and on. I periodically squash the filters every couple of weeks or so and pump away the resulting dirty water out the system (google oase filtoclear 6000). I have a number of maintenance tasks I do but to extrapolate these to yours - but I'd like someone with a biologic skippy to chime in please as it's not the same. That said the processes are really the same.
My filters get clogged and saturated with waste and replacing them means that the water almost immediately clears up, because there is now a way in the system to capture more waste mechanically. I believe that the equivalent on your system maybe something called backflushing, ie stirring the sponges in the filter and flushing any accumulated goop on the bottom. As you have two, the first is designed to have most of the goop settle at the bottom to be processed by nitrobacter followed by the second which should tidy up the remainder. Looking at the green water they're both a bit overworked. see here -
https://www.gardenpondforum.com/threads/how-do-you-backflush-a-skippy-biofilter-with-the-drain.7587/ - so it's easy, switch off the pump, stir the media in both and release the goop in the tanks, make it dirty and smelly ie get a lot of badstuff released into the tank water, then flush this now dirty tankwater by having the outlet hose not run to the waterflow but releasing it
out of the system instead ie into the garden, then once the water coming out is clear again reconnect the whole flow back into the pond ie back to the waterfall. The only bit I'm unclear about is media, hence my question at the start. As in these Skippy filters seem to describe various filter systems, some just biologic (ie just plants), others mechanical too (with filter pads). I don't know which variant you've inherited, or if adding a bunch of filter media like pads or whatever to medialess tubs helps - others should chime in on this as I'd rather not misinform.
It's interesting that nature on its own does perfectly good filtration systems that filter the water well enough that stuff doesn't need to be flushed away. For example crazy as it sounds, you could probably fill your entire smaller goldfish pond with reeds and run the water through that rather than the skippy filter, and that'd probably trap the waste and be a better filter for the larger pond than the manmade skippy. Some people on this forum have awesome large natural filtration systems like this for their very large projects.
Obviously afterwards you'll want to refill a bit and dechlorinated water is preferable to tap water - there are chems available to do this. In effect this'll be like a small 5%-10% water change anyway - these are good, because they reduce the amount of contaminated water. You can then also reseed the filter media with beneficial bacteria if you want to speed up the cycle again, and you can buy more nitrite consuming plants into the skippy filter like watercress or the very efficient water hyacinth that you already have so you have more of a cleaning engine going on.
One thing, the pump - the unit in the water that pushes the water to the filters - can itself also get clogged and washing this out so that the pump's draw (suction) encompasses the whole pond again is also a sensible thing to do. Because a pump that doesn't pull in waste from the whole pond to be nitrobactered isn't doing what you want it to. So whilst you're at it getting dirty and everything is switched off, pull up the pump, scrub the inlets, hose it off and clean obstructions so that the inflow is nice and clear again, then put it back. Fyi the yardstick is that a pump should be able to rotate the water every 2 hrs or so in your pond, if not more.
Once you've done these you should start seeing some improvements although if you still have a lot of accumulated gunk at the bottom of the pond (as opposed to inside the skippy filter tubs that you already cleaned) that isn't even reaching the filters then you'd be best off getting that out as well, net by net, as others have suggested. Slow and easy with a fine mesh net should do it. Or suck it out with a pondvac, but that might be pricier than you want to spend. The more of that you get out the lower the percentage of goop you have in the pond and your nutrient levels should start to drop and that will in turn starve the algae. But, if you want to kill the algae on a continuous basis, a UV light somewhere in the system will definitely help, and that'd be worth looking at adding to the loop, maybe go to a local koi shop and get their view on which is best wattage wise and where to place it. Bring along details of the pump though, the model number, because its speed will help determine the best wattage UV unit.
But with cleaned out filters, a cleaned out pump, cleaned out bottom, new UV light and more natural biofilter plants, I'd be surprised if your pond doesn't start turning nice and clear within a couple of weeks or so.
