Hi, I'd like to help you with your question.
It is very hard for children to understand why adults act the way they do. When your niece reacts to her father's rejection, she may be feeling that it has to with her. Children can interpret a parent's rejection as there is something wrong with them.
What you or your sister can do is talk with your niece and explain to her that her father has issues that he needs to deal with but that his behavior has nothing to do with her. Explain that sometimes adults make poor choices or can be confused about themselves and their lives. When they feel that way, they may not make the right choices. Avoid saying anything derogatory about the absent father, even though it is tempting. This would only serve to make your niece feel bad and possibly angry at you or her mother. She may also become defensive about her father and begin to blame you or her mother for her father leaving (if you weren't so mad at him, he wouldn't have left). By keeping your tone and comments neutral, you help your niece make her own conclusions.
Also, emphasize with both children that they are never at fault for what an adult chooses to do. Let the children mourn and express their feelings as much as they need to. Missing school for a few days is fine but encourage the kids to return as soon as they can. This helps them learn to cope and keeps them from focusing on how bad they feel.
Also, allow the kids to ask questions. The more open the communication, the easier it will be for them to heal.
If the kids' father refuses to see them, let him know that the offer stays open but discontinue setting up appointments with him unless court ordered. The disappointment to the kids is too traumatizing because each time their father cancels, the wounds from his rejection are reopened. If he is interested in seeing them, then he can make the effort to do so.
The best thing you can do is to be there for the kids. You may not be able to help them monetarily, but offering to take the kids for a day so your sister has time to work out her issues is an enormous help. It also gives the kids time with another adult who cares for them. Take them for ice cream, watch a movie together, create fun times. The more good memories you can provide for them, the less trauma they will feel.
I hope this has helped you,Kate