For me the only, and do mean only, thing that kept me going was my children.
There were days when I didn't get out of bed (or off the couch). There were times I didn't shower for nearly a week. I didn't pay bills. I didn't cash paychecks. I ended up having to take a leave from both my part time jobs because I would just start crying for no apparent reason while working. I was depressed and didn't see any way out of the problems I was experiencing...except death.
Knowing that my kids needed me was the only thing that kept me going. I just did what I had to do. I went to their stuff (although a great deal of the time I was only physically present). My youngest was around the age of your daughter at the time I was dealing with this. I did not spend alot of time doing extra things with them. I did the minimum. We talk about that time sometimes now. They know I was sick.
I had a couple friends who made me go do stuff. They made me get out of the house. One took me around to pharmacies and helped me (because I really was imobilized)look up info on meds.
Going on medication helped me dramatically. If you are not on meds I stongly suggest you talk to a physician or a mental health professional. I also did counseling. The combination of the two was very helpful.
I'm not sure if I have effectively addressed your question about fuctioning as a mom while you are depressed. You are obviously very concerned about how this is affecting your parenting and your daughter.Try to just keep that focus and literally make yourself do stuff with her (I know this is incredibly hard). Get help...you will feel better.