Shroomjohn,
I'm a retired cop. So far you've done everything right. It sounds like you were respectful, honest and sincere.
At this point I think a lawyer would be a waste of money. You know you are guilty, the cop knows that you are guilty, and you told all of us that you are guilty. Set up a court date and get a copy of your little girls medical records from the hospital. Try to talk to the officer at court before your hearing {preferably in the hallway outside of court}.
We deal with so many @$$holes, he will proably respect you and try to help as much as he can. Possibly a partial fine and a conviction held "under advisement" pending no more tickets during a specified period.
If he happens to be a jerk (although from your story I doubt it), All you have to do is stand up in court and say, "Your honor I request an adjurnment so that I may hire a lawyer." NO judge in America would convict someone without a lawyer who is requesting one. The conviction would be overturned, and they would look like a fool!
Try it yourself first.............Good Luck................Art.