I would suggest that you speak to this awful child's mother for a start. Never invite her to your home again. Then speak to the teacher so she can monitor the behaviour and encourage your daughter to find new friends for you to invite over. Good luck.
young girls can be evil!! I second dippy about speaking to the mother and to the teachers so that they are aware of it and definitely stop facilitating the friendship outside of school as this awful brat will keep seeing it as a reward.
As for your daughter's DF - what an utter shit to forget her birthday! No wonder the poor child is distraught at school - hope she is ok OP