first of all about kawasaki disease, the cause is unknown, it is unpreventable. It is most common among children of korean and japanese descent. In fact there isn't even a test to detect kawasaki disease, which tells you just how rare it is. So essentially no one knows what causes it, no one knows how to prevent it, and it is extremely rare.
As for the chemicals being dangerous, that just is not true. 25-30 years ago, the chemicals were fairly dangerous, mostly to the carpet cleaner, because of long term exposure. Nowadays the chemicals have drastically changed. I am a carpet cleaner. The primary chemical I use is incredibly safe. You can drink it, (it tastes like soap). This in fact is common with modern carpet cleaning chemicals.
As for excema and allegies, quite frankly cleaning the carpet will almost definately help with that.
My chemicals are also all natural and bio-degradable, and quite frankly if the carpets are cleaned properly there will be no chemicals left in the carpet. My chemicals by the way evapourate completely at room temperature in about 30 minutes. This means if I somehow miss cleaning some spot teh chemical will just go away on its own.
If you want to find out if your local carpet cleaner has the same sort call and ask.
I recomend a truck mount unit with a rotery cleaning machine. The cleaner should also use the prespray method.
Some other info that might come in handy, check and see if your landlord will pay for professional cleaning. A landlord can claim it on there taxes, which is a good selling point