Oh, you are living my nightmare! I am personally allergic to dogs, but I control it because I love them too much to give them up. With 3 dogs (and being actively involved in animal rescue, I know how awful it is for dogs to be turned in, even with a legit reason), I am terrified that my son will have allergies and not be able to take the meds I take to deal with them! I am very glad you decided to keep your dog. I would, too.
Additional advice that I can offer - make sure your vacuum is bagged (bagless vacuums are convenient but no good for those of us with indoor allergies) and has a hepa filter.
If your child has carpet in her bedroom, consider (if you can afford it) switching to hardwoods, which don't trap allergens as easily.
Clean your furnace filter more frequently, about once a month, and make sure you don't buy the super-cheap filters because those don't catch anything. When you get your furnace serviced, ask the guy about what sort of filters will work well in your furnace and help trap allergens (don't let him sell you anything, though).
Your dog should not be allowed in your child's bedroom, of course.
Depending on the kind of dog you have, you might consider shaving. This is not a good solution for every breed, and of course, dog skin also causes dander and so it won't eliminate the allergens. But if you have a breed of dog (as I do) where the fur seems to hang in the air after being shed, shaving may help reduce the amount of dander in the air. Please remember that if you shave a dog, your dog becomes particularly at risk for sun burns and will need to wear sunscreen just like a person.
In the winter, put a cool-mist humidifier in your daughter's room (if you can afford it, the best ones are the ones that tell you the humidity in her room at all times so that you do not over-humidify and risk mold or mildew) and run it when she sleeps. In the summer, keep your daughter's bedroom window closed to prevent spores of allergens from coming in.
Finally, invest in a neti pot (or because she is so small, consider a neti sprayer). These take some getting used to, can be slightly painful in the beginning (especially if your sinuses are tender), and your daughter won't like it at first. However, a Neti pot takes salt water (or saline) and rinses out the nasal cavities and sinuses. Once you get used to it, it stops hurting, and provides relief. It also clears out a lot of the dander and other allergens that get stuck up there and cause infections. You may want to ask your ped. or allergist about one for such a small little girl. This is a surprisingly helpful step, and was how I was able to survive living in my house with 3 dogs while I was breastfeeding and could not take my normal allergy medications.
I wish you the best of luck!