Public schools do not grant the I-20s required for F-1 visas. If she were here as a dependent such as H-4 or J-2, she would be admitted to public schools. The problem is obtaining a visa for entry which requires the I-20 for a student visa.
Does your son have the education and skills for an H-1B, L-1, or J-1 visa? Would he be interested in an F-1 visa for himself? Could you support his tuition and the living expenses for all four of them? If he were the primary, the children could enter as his dependents.
I am only familiar with the "kinship care" term in relation to foster care. As a foreign national living abroad, she would be ineligible for foster care. You may be thinking of the children who entered illegally and had to be placed and it was done with extended family members just to get them out of holding facilities.
As for local attorneys, be sure they have the proper credentials, experience, and contacts in the Ukraine.

