If your child had W2, her employer and her will be paying total 13.5% payroll tax, that's 50% by her employer and 50% by the employee. This is the same as your W2 situation.
Since your child has no W2 or tax withholding, she can claim this income as self-employment. She pays 13.5% self-employment tax (because she's both employer and employee, i.e. self-employment), then this is her earned income and she can contribute to IRA.