When I taught preschool I got sick a lot. I used to want to blame the parents who send their sick children to school. But the truth was when people are the most contagious is when they first get sick and have barely started developing symptoms. So the child who has had a cough for a week and a runny nose is not the one infecting everyone it is the one in the corner that is about to crash that is spreading more sick germs. So you can see how it is a vicious cycle. Sanitize the toys on a regular, work with your DC on better hand washing like you said, and covering coughs and sneezes with elbows!
Good luck

it will get better, it might take a year to build up DC'S immunity (it took me much longer as an adult working in that setting!)