Start with the Payment Assistance Application (at least ask about it - both with the hospital and with the lab).
If that doesn't work, then you are going to need to start going through the actual paperwork/contracts that you (or your wife) signed when she received her services (at the time of check in with the hospital). This is going to give you the legal obligations for both you and the hospital (as well as third party providers) for billing and payment responsibilities.
Unfortunately the ultimate payment obligation is going to fall on you and your wife, but do read the contract carefully, if you can identify that the hospital was supposed to utilize a different provider and failed to do so, you can approach them with the specific clause identified, and negotiate their responsibility for this payment.
If you find that you are ultimately responsible for the bill, and there is no other responsible party, then you can try negotiating a reduction in the bill with the provider directly.
When trying to settle a debt, creditors generally prefer lump sums over payment plans. They are often willing to accept an amount less than the full debt (the trade off is that they get a quick payment and don't have to worry about ongoing collection costs or administration). If you do not have the ability to offer a lump sum for something the creditor will accept (some will accept a small portion, while others want close to the full amount), you can try a payment plan, these are less satisfactory to the creditor (especially if they have a lien on your property already), but if you are willing to offer something with a reasonable chance to get the creditor a large amount of their debt back, you are likely to get them to accept it.
Whenever working with a creditor, make sure that you keep your communications in writing (if you speak to someone by phone, promptly send a confirmation letter to summarize your conversation), as this will help to ensure that there is no confusion later on, and you will be able to enforce your settlement against any future collection efforts.