You made the promise. Your niece didn't come to you; you went to her with the trip. If you go back on your word no matter what the reason, I would never trust you again. I don't think it's fair to your own daughter, either. You stated this was a birthday situation and that they are very close to each other. I would be insulted if you went anyway and you left me behind.
I'm not understanding over $700 for tickets for your niece alone. This sounds outrageous. A Disney annual pass doesn't even cost this much. If this includes other tickets, then cut back. If you already purchased OOP all the tickets and can't return them, then you'll have to cut back on other costs.
If you go now then cut back on costs. This may mean taking a shorter trip, cheaper dining, cheaper resort, etc. Take a trip to the grocery store in Florida and eat in your room for some or all your meals. If you get free dining then this might not help, here. Change your dates to see if you can get a better deal that way. Maybe, you can get in on a better promotion that way. I would use an airbed, sleeping bag or the equivalent and not pay for a bigger place, etc. Maybe, consider driving back and forth if it saves more.
If the above doesn't work for you then postpone the trip. I don't feel your niece or her family is responsible for incurring any costs, here. Where's your Disney magic? Sacrifice some and I think you can make this workable to include her. I'm sorry you are sounding selfish to me. Give up some on your part and it will work out.