I asked quite a few questions in my response to your PM about this. Feel free to answer them here.
As you discovered, vacuuming disturbs the biofiltration contained in the Live Sand. Nitrogen pollution becomes noticeable if the filtration cannot handle all that the tank is producing. In a way, food is the culprit. Assuming that Ammonia and Nitrite were not tested, you can be sure they have risen too.
The stuff floating around in the water is not harmful. In actual fact that stuff is mostly clumps of bacteria that have been stirred out of the sand. I wouldn't worry about it too much. It will settle.
Feeding less is the best answer in the short run. Developing more biofiltration capacity is the best long term fix. Evidently there was more than one biowheel running. Removing all at the same time would not have been advisable. We give the same advice with regards to bioballs; removed slowly in steps. The reason for this is that it takes time for biofiltration to catch up to a sudden change. The way I see it, the problem with using biowheels is that if something happens to that electrical unit, there is nothing to replace it. We have also seen that biowheels actually retard the development of an efficient biofiltration within the tank. In my opinion and experience, it's better to develop the tanks biofiltration to take care of the pollution. This is also why I have been heard to say that a Skimmer is not a necessary part of reefkeeping, because the tank can become overdependent on it.
Anyway, there's no sense crying over spilled milk, now that the biowheels are gone, the issue is what to do now

As an important note, a temporary spike in N compounds will not seriously damage the animals. Many tanks run just fine at Nitrate levels above 20 ppm and Nitrate levels approaching 50 ppm is not necessarily a serious emergency.
Best thing to do right now is to stop feeding completely. This will allow N to drop quickly as the tanks biofiltration catches up and processes the current waste. I look forward to the answers to my questions so we can help you improve the situation with a long term fix.
Aloha,
Mark

P.S. Fish and coral will be just fine without feeding for a week or so. They actually appreciate the improved water quality that quickly cleans up and feels fresher when feeding is suspended. The fish then eat what they find in the tank. This makes the entire tank look cleaner.

Also, as you have noticed, large water changes are not the answer. A 25% water change may reduce the pollution level by 25% for a few hours or a day but if nothing else is changed/improved the pollution levels often rebound to previously unacceptable numbers.