You are the one that sought care. The responsibility for the bill is yours. It is unlikely that your own insurance would cover it if it was not submitted within 1 year. Statute of limitations is 6 years. If you have not made a payment in 6 years - they cannot collect unless you make a payment.
The daycare might have some liability with respect to a tort. ie.. negligence. THe statute of limitations for that is 3 years - however, it is tolled for a minor ie... a kid. Contact a local personal injury attorney. Also, post in personal injury section.
Yes, based on the facts that you set forth it is highly unlikely that you would still be able to go after the daycare to pay this bill. You have waited too long before pursuing any kind of negligence claim against them.
Once six years passes from the date you last made a payment to the hospital, or the date services were rendered, whichever date is latest, then they cannot sue you for the debt. This does not mean they can't continue to contact you about paying it, but it does mean they would not be able to use the legal process for things like garnishments.
You did get the services however; if you have the ability then you should contact them about payment arrangements. If you are not able, and you fear they will sue you before six years passes, then consult with a bankruptcy attorney.
Best of luck,
Tim L. Eblen