What should the questioner do with the extra amount that was deposited into her mother's social security account, knowing that she is employed and not eligible for Zakat, and the amount was increased because her name is included with her mother's in the family card?

The amount deposited into the questioner's account, being her share of zakat (definition), must be returned because she is an employed individual who is not in need. It is not permissible for someone who is not eligible for zakat to benefit from it. If the funds are not from zakat, then reference should be made to the conditions of the granting entity. If their conditions stipulate that an employed person who is not in need cannot benefit, then the aforementioned amount must be returned. The fact that the employee was aware of your situation when the documents were submitted does not permit you to benefit from this money, unless he was authorized to do so by the competent authorities, or if the conditions allowed it.