Otherwise, you're probably SOL. Could try taking her to small claims court, though.
(I like Airbnb but see how fraught it is, too).
It's more like the your renter's insurance company would sue the neighbor's company for a claim. However, since the neighbor was renting out the apartment as an Airbnb property, then their standard renter's insurance was probably invalid at the time. So now your insurance company would sue the neighbor directly and may not be able to pay.
The long term implications are that renter's insurance costs will go up for everyone in that market.
I've long thought that the biggest problem with Airbnb wasn't the tax issue, or even landlords abusing the system and running mini-hotels. It was insurance. I suspect that none of the hosts who aren't physically present (and even some who are) are violating their insurance policy, and if there is a HOA or other sort of shared space, that insurance policy would also be violated.
If hosts had to actually pay the true costs of insurance, the cost savings between traditional hotels and Airbnb would be much smaller. Now, if Airbnb actually offered the insurance to the hosts, or maintained it themselves, that would be something. But that would be fraught with fraud issues over such a wide market.
But insurance isn't a sexy topic...
The cost to the aggrieved neighbor of launching multiple legal actions is proportionately greater since the neighbor's recovery necessarily takes place in arrears and is not supported by a stream of guest rental income. Absent an ability to recover punitive as well as economic damages, the rentier may well find it more profitable to persist.
This leads to an interesting application of the Coase theorem, of course. If you find that your neighbor is renting out on AirBnB and you have a serious anticipation of economic loss, you are in a good position to bring your neighbor to the negotiation table and solicit compensation up front - in other words, to demand your neighbor buy you off for the predictable inconvenience. This idea will seem anathema to many entrepreneurs but it has the economic advantage of predictability, as the long-term economic cost will be the same while the transaction costs may end up being lower.