Pneumonia may be a possibility.
I would request a chest x-ray from your M.D.
Also, get a prescription cough medicine if possible.
They are much more effective than otc.
I second the sleep w/a humidifier @night suggestion.
If you do not have one, perhaps your bosses have one you can borrow until your coughing subsides.
Lots of liquids + rest as well.
And napping while the child naps may also be very beneficial for your recovery.
I cannot imagine any employer having any issue w/it as long as you are near the child or baby monitor.
And especially since their child actually made you sick.
Get well soon OP!!