Carpeting can be a reservoir for allergy-causing substances (allergens) that trigger asthma. Carpeting in the bedroom can be especially problematic because it exposes you to carpet dust throughout the night. Hard-surface flooring such as vinyl, tile or wood is much easier to keep free of dust mites, pollen, pet dander and other allergens.
Steam cleaning carpet on a regular basis can help reduce the presence of dust mites and other allergens in your home. If that isn't enough, replacing carpeting with hard flooring may be a good idea.
If you do put in hard flooring, keep in mind that all synthetic flooring initially releases gasses known as volatile organic compounds (VOCs) that can worsen asthma. In most cases, products stop releasing VOCs after several days. However, some products release more VOCs than others do. You may want to look into low-VOC flooring if this is a concern.
It might be worthwhile having your daughter see an allergist for allergy skin testing. There may be individualized steps you can take to reduce allergen exposure once you know exactly what your daughter is allergic to.