Well that sucks! It is tough to lose 3 families and then have the one remaining asking for a discount!
I would definitely change your contract and have her sign the new one. My feeling is that she will never be the client she once was as she will not like you not giving her a discount and having her sign a new contract. I am sure she is already looking for a new daycare. Personally, I find her unreasonable and would not be keen on keeping her on. Six minutes extra minutes is nothing when you know your child is well cared for and happy.
Good luck with your move and welcome to the forum