Did you pay her above board - ie pay unemployment taxes? If so, she might file for unemployment but you wouldn’t foot the bill - it would come out of the taxes you already paid.
(I believe your unemployment tax rate will go up if she files - of course, this only matters if you hire a new nanny in the future and pay her above board)
Just to be safe, I’d document her last day, reason for termination, etc.
This may sound inappropriate and if so please disregard but… your nanny sounds like she is taking advantage of you. You are not responsible for her future. You have no lasting obligation to her. I think you are understandably grateful to her and feel a connection with her, but you’ve held up your end of the bargain - gainful employment and a salary. It’s her responsibility to decide on her next step.