Since you appear to be currently offline, I will leave my thoughts here.
The signs you have described are all very suggestive of stress + oxygen deprivation for the fish. We can see similar signs with nitrite or nitrate poisoning; but if the pond has just been cleaned out this would be less likely. As well, I assume that the pond cleaners didn’t use any cleanser that may have left a toxic residue here (and hopefully the untreated hose water is not grossly abnormal).
Now stress in fish can arise from any sudden change in their environment. If the water is completely new, then this can be the source of the problem. We can see problems with total water changes due to temperature fluctuations, pH shock, and water hardness alternations. Koi are particularly sensitive to rapid temperature and pH changes. As well, we can even have situations where the previous pond water was terribly toxic (with high nitrates/nitrites/ammonia); yet when we change it to new water we end up with fish that cannot cope with the dramatic change. This is why in nitrogenous waste toxicity issues, we have to do gradual water changes to wean them down to normal, clean water without inducing fatal shock to their systems.
As well, we need to consider the oxygenation ability of these fish, especially if you think they are struggling to breathe. While fresh water should have a good oxygen supply, it is possible that the stress of change has led to the fish having an elevation in respiration and this then lowers the oxygen resources of the water. As well, if the stocking density in this pond was at maximum for the pond’s oxygen carrying capacity (ie ideal water volume is 100 - 150 gallons per small koi (2" - 8")), we could see this problem be secondary to the lower volume of water present now and thus lower volume of oxygen. As well, if the oxygen balance in the pond was reliant on the filters or any supplementary aeration, these being off could also play a role here.
In this situation, it would be ideal to check water temperature and water parameters (ammonia, nitrites, nitrates, pH, hardness) and it’d be even better if we had pre-change levels to compare it to (to see if there was active nitrogenous toxicity present that has led to a system shock). If anything is amiss, then it needs addressing. Since the ideal water temperature for koi is between 65 F-75F, you do want to make sure thew water is within this range. If it is not and you need to raise/lower it, do consider keeping to the lower end of ideal for them since they will have a lower body oxygen demand in the cooler temperatures (which may just give you a chance to get the oxygen angle sorted). Otherwise, you do want to take steps to make sure there is adequate oxygen in the water for these fish. This means turning the filters back on (as you have) and potentially adding further oxygen supplementation (ie add airstones, etc).
Overall, it sounds like these are fish in shock and potentially under the stress of compromised oxygen availability (likely related to the volume of fish struggling to cope with both the severe environmental change and the reduced water/oxygen access during the refill). Therefore, do check the above (or have the experts do so) and make the appropriate adjustments to restore the pond's default state. Hopefully, once everything is back online and the standard pond set-up restored your fish will be able to settle.
I hope this information is helpful.
If you need any additional information, do not hesitate to ask!
Remember that if you have any lingering questions or concerns, please reply so that we may continue our conversation. I will be happy to work with your further and do everything I can to provide you with the service you seek. Please remember to rate my answer when you are satisfied (with 4-5 stars or a happy face) so that I may receive credit for my assistance. Thank you & have a great day. : )