First, I would double-check the written terms and conditions of your travel package (and the terms and conditions of your airfare), as well as your travel insurance, and see precisely what they say about non-refundable airline ticket costs in the event of cancellation. (If the package terms and conditions are the same as those Disney posted online in 2015, they do clearly provide that: "If air transportation has been ticketed, guest will be responsible for any increase in airfare as well as the change fee assessed by the airline." ) Still, double-check, and re-read the terms of your insurance coverage to make sure that your particular "family circumstances" aren't a "Covered Reason" for cancellation. (If it's a medical reason related to someone in your traveling party or immediate family, they may be -- otherwise, probably not.)
If the terms of your package provide that you are responsible for airline cancellation fees, and that the Disney travel agency doesn't absorb any of those fees (or won't agree to do so -- it wouldn't hurt to ask especially if your circumstances are ones to which they might be sympathetic), and that your travel insurance doesn't cover it, then you are stuck. You'll either have to pay the fees, or call Delta (asking for a manager with the power to waive your fee), explain the situation, and throw yourself on their mercy. If that doesn't get you anywhere (and candidly, it's not likely to), posting about it on Delta's Facebook page or Twitter account (in a credible, reasonable post - not a rant) might draw some attention as well.
I'm sorry that you're in this situation -- unfortunately, airlines charge these ridiculous fees simply because they can get away with it. For your next trip, I suggest either purchasing "cancel for any reason" travel insurance or waiting until you are absolutely sure of your travel dates to purchase non-refundable components of your vacation, like airfare. Also, if you can help it (and this comes from my family's own personal experience with the airline, which utterly ruined a vacation my parents had planned, scrimped and saved for for years, costing them thousands -- in recompense for which Delta offered them each a T-shirt and $100, I kid you not), never, ever, EVER fly Delta!!!