Legally, the children are qualifying children of your's. However, they are also qualifying children of their mother. If your niece has no income, she cannot claim her children because she will not file taxes. She may ask someone else to claim them (illegally) in order to get them to give her the excess money.
You would ultimately need to prove to the IRS, if the children are claimed by you and another taxpayer, that the children lived with you and you provided more than 50% of their support.
Report cards for the children addressed to your house would be helpful. Any type of school records for the children with your address would be helpful.
Medical helps prove your support as well - do you have any receipts for doctors, dentists, or prescriptions for the children?
Receipts for necessities for the children - clothing, additional food, etc. - will not stand alone as the only proof. They will help along with the other receipts to prove you provided at least 50% of the support of the children.
I suggest you try to reconcile with your niece and attempt to create a notarized letter from her that the children lived with you all year and you provided the majority of their support. That would be your best proof to provide to the IRS.
Please let me know if I can help you further.