Something in my home is making my family sick. How do I determine what it is?
There could be several factors that may be contributing to the situation. It sounds like an air quality problem. This could be caused by airborne mold spores, chemical contamination from bad paint or flooring adhesives (VOC's) etc. , CO from a malfunctioning heating or A/C unit or something from outside getting in. I have had to address this problem with some customers in the past. We contracted an air quality testing from a local environmental testing lab. In your case, I would first consult with the management of the property. Explain the situation, confirm your conversations in writing, (document, document, document) and see if they will contract this service. If they won't for some reason, then I would have it done privately. Depending on the results and plan for mitigation, these costs should be reimbursed by the landlord. At minimum, it would give just cause to break the lease. If the results are bad and the management won't work with you, share the results with your local code enforcement office as they may help you and require the management to fix the problem.
Once the type of irritant is identified, then the cause will be much easier to find.  Good Luck.