4yo is on augmentin (amoxicillin and clavulanate) day 5 for sinus infection double ear infection. and she's still has a juicy bad chronic cough. and junky nose. when will this end?

Sinus infections: often take longer to go away than 5 days. Sometimes they need 2 courses of an antibiotic. Your daughter may also have a concurrent viral infection causing her cough, which certainly can last longer than 5 days. If she is not better after 7-10 days on her medication, or if she is worsening, she should be re-checked by her doctor.