Can I include the fees?
No. That would normally be a miscellaneous itemized deduction subject to the 2% rule, but the 2017 tax reform law eliminated that deduction for 2018-2025. You report the gross wages, before any deductions for the nanny tax, but not including service charges from your payroll processor.
I don't understand the rest of your question, the sections you are asking about don't seem to match either schedule H or form 2441. When completing the child care credit, you are expected to know the amounts you paid for the caregiver. Turbotax doesn't link form 2441 and schedule H because you might have more than one household employee, or your employee may provide more services than child care (which would not be eligible for the credit), so Turbotax can't assume that 100% of your schedule H wages are applicable to child care.
