I personally wouldn't drop the day sleep as you said she seems tired during the day and it affects her behaviour. But I would just make it shorter instead and see how she responds.
My DS is same age and really needs his day sleep. But sometimes has phases of taking ages to fall asleep at night, for no obvious reason. We just ignore him or if he's being loud go in and say go to sleep please then leave. Its an attention thing for him when he does it.