Baby with animal allergies!?(sorry long)?

[Soquel G]

My daughter is 10 months old..And has been raised around animals sense day 7 (she didn't get out of the hospital til then) And like usually happens she didn't have any issues..I am a HUGE animal lover and we have 4 dogs,1 of which has 9 pups and a cat that lives with my mom (she doesn't want to give him up) but my daughter lived with him from 5-7 mnths..also her other grandma has 3 cats and 4 dogs...(we also have a rat and lizard but she never comes in contact with them..

Well recently I brought home a lil black kitten..And a few days later my daughter had a runny nose, rashy cheeks,and a lot of eye gunk..It got so bad in the morning her eye lashes would get stuck together..

So I brought her to the Dr..who agreed that she may be allergic to the kitten and prescribed benadryl..it seems to help a bit but she still has a lil eye goop and lots of boogers..
The Doc said give it 4 to 6 weeks to see if she get immune..
I always knew this was a possibility as daddy is allergic of dogs and cats..(cats more..along with peanuts,grass and so on..)Well iof I have to get rid of this kitten I will but im not geting rid of the cat at my moms house..I love him too much and have had him sense he was a kitten 6 yrs ago..if anything he will be an ouitside cat or be allowed in only our large laundry room..he will have to move in with me when my mom moves in with her BF in the next yr..

Does anyone have an allergic baby and animals? What helps?