First, I freely admit I am NOT a professional, but I was taught back in the girl scout camp days that ticks burrow in clockwise, so if you grasp the tick as close to the skin as you can, and give it a gently twist/turn counterclockwise before you pull (gently! you don't want to rip the body off!), it makes the pincers release so when you pull, you get the head, too. I've been doing this most of my life now, and have yet to leave the head in. (I became the official tick remover of the family @ about the age of 9 as my mom is bug-phobic!) Once the tick is out, check out it's little head, if it's got a bit of skin attached, you got the head. I've heard the smothering thing too, and as far as I know it works, I'm just too stubborn to let it sit in me longer than it absolutely has to! I've also heard you can light a match (the big wooden ones) blow it out, and apply the hot tip to the tick to make it reverse, never tried it, tho! Obiviously as the other moms have said, cleanse the area afterwards! Also, the less you freak out about it, the less your daughter will freak out. Like I said my mom is completely bug-phobic and thank god my dad was laid back about it and the camp counselors were, too, otherwise I'd be as chicken as my mom!
I would read up on lyme disease, too, just to be safe, and learn what to watch for for symptoms. I wouldn't stress too much about it tho, especially if the tick is bigger than the head of a pin and has either a white necklace marking or white racing stripe markings on it's back. Those are the normal ticks that have been plaugeing us for forever!