OP, I'm sorry to hear about your baby. I hope your little one gets well soon. You sound understandably distraught and stressed about the whole situation. You're a good mother to worry about your kids' privacy being invaded in a scary way, and a good wife to worry about your husband's feelings about his family.
Personally, in light of the issues you've touched on, I would drop the rope solely so I could invest that emotional space elsewhere (on my sick child). You do have the right to be upset that someone didn't follow through with an agreement, but as I mentioned to another poster earlier, if you feel able, try to give up that right. You really need to focus on yourself and your baby right now, and dealing with issues outside of your nuclear family isn't really possible with everything else pressing on you.
Perhaps you could try letting your husband handle things with his family. Sometimes that means things like this get let go when we would rather them not, but let that be his choice to make. He is dealing with it, just not in the way you would like. He's choosing to let it slide. If you can, try to support his decision, as it sounds like you need to be kind to yourself and not take on something like this when you're carrying so much already. 
I understand that I don't have all the details, so if this response isn't helpful, please know that I'm wishing you the best, and a speedy recovery for your little one.