J.G.
I don't know what the regulations are where you live. Here a landlord must keep the enviroment below a specific level of mold. You just have to call, well I don't know who, the department that regulates such things. They will test and if it is at high enough levels they will compel the landlord to fix the problem.
Having said that my house can have mold at fairly low levels and it still gives me headaches and sinus infections. Severe allergic reactions will cause sinus infections. I found that out when I went to NYC and found I was allergic to the whole city. :( My point is that you can have enough mold to cause your children issues but not enough to compel your landlord to do anything about it.
Here is my two cents if your husband is determined to fix the leaks. Go up in the attic, look for the water marks. Find the top of the mark and that will give you some idea where the roof is leaking. Remeber that water takes the path of least resistance. In other words there is no water with an agenda going up under the shingles. :p Look for missing or cracked shingles also check the flashings around the vents. Also on the vents check to see if they have black gaskets around the vent pipe. Those gaskets can dry and crack over time and leak. If you have to replace the flashings around the vent pipes do it on a hot day when the shingles will be softer and easier to bend without breaking them.
Worst case hire a roofer and send your landlord the bill. Just be aware he may not promptly pay it.