The left and right hand will almost never know what the other is doing in these cases. Billing in one area isn't likely to communicate with payroll or benefits.

Some hospitals won't bill insurance after a certain point, and insurance sometimes has a time limit on how long they'll accept a claim. It's still worth pursuing as an opportunity to get it paid off.

Get as much information about your account down as possible. The collections process varies a lot depending on state and the company you're working with, often the creditor (Your hospital) will not communicate with an account that is collections without the involvement of the collection agency, but given that you work for them you're in a better position than most to get those conversations going.

It's worth saying collections isn't the nightmare beast it used to be due to a lot of consumer protection laws. It's never a lost cause to communicate with the collection agency, your insurance, and the hospital to work something out. Even if that's just to set a payment arrangement and keep it off the credit report as unpaid.