I agree with most of what Harlock said. Water changes are what nature uses. I would do one thing different: wait at least 1 hour between water changes. I do this for a couple of reasons:
1) to make sure temperature doesn't fluctuate too much (of coarse you could add same temp water, but not very exact) the fish are already stressed and don't need to be chilled (cause ich or other disease) or over-heated now (which causes increased respiration/lower oxygen levels).
2) your water could have a high amount of carbon dioxide (from the pipes) and needs to adjust to ambient levels. This increased CO2 could cause a pH drop which could, in turn, cause increased stress, disease, or shock.
3) with such a rapid alteration of water chemistry, your bacterial colonies (used for biological filtration) could be shocked or killed off. This would probably be to minor degrees, but at this point the cause of your problem is not having enough of these beneficial bacteria.
50% water changes sound good to me, but I wouldn't go over that! Within an hour things should stabilize a bit, then I'd do another 50%, without bothering to test. I wouldn't test because if you are having high amounts you need to get nitrites as low as possible. After the second I'd test. If it's higher than your lowest detectable amount (whichever is just above 0) I'd do a third 50% in an hour. I'd consider doing daily or every other day water changes to keep this low.
My testkit is a Aquarium Pharmaceuticals Nitrite Kit. It has increments of:
0 ppm
0.25 ppm
0.50 ppm
1.0 ppm
.....etc.....etc....
If it ever got to 0.50 ppm, I'd do a 50% water change. No mater how often! It may take 2 weeks of this Nitrite spiking (sometimes more), so just be very careful and pay attention to your fish. Be persistent and you will reap rewards! God bless!