Uhhh, if someone asked me to "justify" my rate? We would probably not be working together!!
IF this family questions you, give them the reasons you listed above. None of them are suspect.
And if someone really wanted to know what their rate paid for, I would break it down for them. List off all the costs that I incur running a home daycare, and show just "how much" money I keep at the end of the day.