Laws vary from state to state, but in most states they have longer than one year. It is very unfortunate that they can wait, but it is actually normal for many offices that are not on top of their billing. It sounds like your insurance paid for most of the bill and this is the balance? I would recommend that you contact your insurance carrier for that time and ask if you had what is called an out of pocket limit. This is the amount that you must pay before all expenses are paid at 100%. Most policies have out of pocket limits around $1500-2000. It is possible that the insurance made payment, and the provider should have made a contractual adjustment based on their contract with the insurance carrier (if they are in network). It is also common for the provider's staff to not enter the information properly in the computer and you are being billed for something that you shouldn't be. I would first contact the insurance company and find out how the claim was processed. Is the provider in network and if so, what was your responsibility for the services. If after contacting your insurance carrier they advise you that the provider can bill for the $3000, then I would contact the provider and ask why you never received a bill prior. If they give you a reason and stick by the bill, then ask if they won't do any form of a discount due to your financial situation and the severity of your baby's care (since she was hospitalized for 3 months).
Start with that and see where you get.
Good luck
Michele