I have to agree that memory would be burned into her brain ... even if Hailey had not gone missing, if for some reason you were questioned about a random day, say March 6th 2004, and you did a lie detector test, wrote your own timeline, spoke about it on tv and so on, you'd remember from going over it so many times. She was lying then or she is lying now.
Now add the life changing nature of those two days. Who drove you to the hospital to have your baby and who took you home again? I'd say there are not many mothers among us who can't retrieve that information pretty easily, and if you can, Billie can remember how she got to work the day her daughter went missing IMO.