@Summer64
Hmmm.. I've had these. In every case, they were actually all staying in the house, using the excuse that she's staying elsewhere. As you can expect no support from Airbnb in these cases, I normally just don't say anything, and write it in their review, while letting them write a glowing review of our place.
But we're not a homeshare, so it's more difficult to get away with it in your case. You might be able to find someone at Airbnb who might listen and do the responsible thing. But that's a long shot.
What @Laura2592 says is correct. Tell them they have to cancel. And failing that, cancel them yourself and take the hit. The only other alternative is to demand more money, but you'll still get a bad review, so cancellation is probably the lesser of two evils.
I wish Airbnb could at least >try< to support hosts when they're being taken advantage of by guests. But apparently it's just not in their DNA anymore.