C.T.
A carpet steamer. Also I have used regular laundry detergent (fragrance and dye free) along with a bit of fabric softener. Vinegar is a great option as well.
That being said... We own a flooring store and my husband says that carpets are not met to be cleaned. The water soaks through to the carpet backing and carpet pad and begins to degrade both. He will only let me clean twice a year and says thats about one time to many. Steaming is the best way to go. If you have a warranty on your carpet it could be void by the manufacturer for cleaning.
I would invest in a steamer. Haan MS30 from HD for $179. It will really help with the asthma.