I sympathize, as I've been struggling with this same battle with my 10 year old female cat. My cat has had issues with peeing/pooping in the house off and on for years, but removing all throw rugs stopped the problem in the past. However, when I was about 8 months pregnant, she resumed peeing in the corners by our front door and nothing stopped her. I used the Feliway plug-ins (pheremone plug-ins to soothe cats) in the past with success, but not this time. I started using non-scented puppy training pads to protect the hardwood floor and make clean-up easier, but that wasn't actually dealing with the problem. It finally stopped when I placed Airwick I-Motion automatic air fresheners in the corners she peed in. The noise and spray scared her away and after a few months of using them regularly, she stopped peeing. Now, we just deal with her pooping by the front door and on my daughter's foam playmat at night. My cats are closed off in our laundry room with a cat door into the garage where their cat boxes and food are during weekdays, because I have a home daycare, so I felt really badly closing her off (and my male cat by default) at night too, but it has been the only solution. I bought a 40" gate which my female can't jump over (she has a bum leg) but my male cat can. She now spends her nights in the laundry room/garage, while my male roams free and still has access for food/litter if he wants it. We left her out last night as a test and woke to poop this morning. While my situation isn't the same as yours, I'm sharing my story anyway as a way to let you know you're not alone. I would try multiple cat odor sprays and deterrent sprays (I had some success with "Dumb Cat") and also try the Feliway plug-ins and spray. The Feliway stuff is expensive, but if it works, it's worth it. You might just have to confine your cat at nights and teach your daughter to close her door during the day. It's not an ideal situation, but it's better than death for the cat.
I WOULD NOT recommend putting a litter box in your daughter's room, as it will just teach the cat that that's an appropriate place to go to the bathroom, on top of the fact that a 4 year old should not be living in a room with a cat box. If that was an acceptable living situation, why not just let the cat continue to poop in her room?
I wish you lots of luck!