She will have to pay the tax owed even if the W-4 was not completed properly. That is because it is always the taxpayer's responsibility to pay their tax obligation, even if someone else was negligent in withholding taxes. It is doubtful the payroll company will pay her tax liability. You do not have any legal grounds for requiring someone else to pay your tax liability since you would have had to pay it anyways if the mistake had not been made, you just would have made the payments as withholdings instead of when the tax return was filed.
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