I like this ^^.
If niece was old enough to go there without daddy, and was old enough to invite friends, then she's old enough to manage those friends and to accept responsibility for the damage that *her invited friends* did to your property and pay for proper repairs. OP, look at it this way: she and her friends saved money by using your cottage for free. Now's the time to take that money they saved and make it right with you. Maybe she needs to save it, but she needs to make it right. She's a grown-up, and she should start acting like one. (refer back to first sentence of this paragraph.)
OP, I think we all get that you want to be generous with your family. But the thing is....they don't appreciate it, and they don't see it as you being generous. They see it as if they're entitled to it, which they're not. So that idea of being generous goes right out the window. What if you'd put the money into a program where you got to go on a cruise once a year? Would you feel the need to give away your yearly cruise just because you had entitled relatives? I hope not.
I like the idea of letting all and sundry know a few ground rules. First, that you have, in good faith, lent the cottage to various relatives. You were trying to be nice and let them use it for free, yet your thank you from them was that people ate all of your food, left your home filthy, and in one case actually damaged your property. Ask the question--point blank--'how is this even remotely okay?' And then some people have started arguing with you about when they think they should use your property. Again, how is this okay?
I would start harassing the niece about the damages. Tell her you want quotes. Or tell her you have a quote and you expect the money by X date. Like I said, if she's old enough to stay there alone, she's old enough to accept responsibility. Welcome to adulthood, honey.
And again, OP, I know you want to be nice, but the thing is that your family has no respect for you or your belongings. Start being a little bit less nice and they'll either leave you alone or 'man up'. And note: when I say less nice, I mean not rolling over and accommodating their every whim. You can be perfectly polite, but you can call them on their entitlement. "You expect me to give up MY cabin for Easter? Yeah, it's mine, and I can go there whenever I want, and I want Easter! That IS why I BOUGHT the cabin...duh....so I could use it whenever *I* wanted to."