To be completely honest I would not renew the problematic mother's contract.
He blatant lack of respect for you and your business is just insulting.
If a parent ever told me they could not come and get their sick child I would call the next emergency contact on the list and they would be sent home. Period.
Now you say she has the nerve to give you attitude ?
You did NOT over react so please don't doubt yourself.
a parent who is only nice to you when things are going well is not acceptable.
In my daycare I do not accept children with a fever.
A noisy,active environment is no placed for an ill child.

