"My neighbor's pit bull jumped our fence into our yard & mauled our family dog so badly animal control had to euthanize him."
Meaning your dog was in your yard and your neighbor's dog was not. Your neighbor is, therefore, responsible for all damages you suffered through the attack on and loss of your dog. However, that is likely to be the replacement cost of the dog and any medical costs involved.
Moreover, I strongly suspect, based upon what facts you've given, that the neighbor is certainly too irresponsible to care about the harm she caused, any lawsuit you bring and payment of the judgment you would receive.
The more productive course of action is to file a complaint with the police, if you have not already, and follow up on it. This includes complaints to your local officials if the PD and/or animal control fail to act on what appears to be a clear and present danger.
The IL and Chicago bar associations can assist you with referrals if you desire counsel.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
Sorry this happened. The Pitbull should have already been taken into custody per the Chicago City Ordinance. You may be able to sue for incurred costs and replacement value of your dog, but perhaps not much else.
I agree with my colleagues that a lawsuit will be counterproductive. Your best course of action might be to file a complaint with the police department and the local village. If necessary, get the media involved. Seek restitution through a criminal or negligence charge filed on your behalf either by the state's attorney or your local village prosecutor. At the very least, your neighbor is more than likely violating a municipal code concerning the permitted limit on the number of dogs.
You might also make a claim on your homeowners insurance, assuming that your neighbor will not furnish you the name of hers. Your carrier most likely will be able to find out the name of your neighbor's carrier.
I am very sorry for your loss.