You do want to make sure that you report the injury to the insurance company as soon as possible. Insurance companies do not want to go to court or pay lawyers’ fees, so most will be able to settle out of court.
However, if they do not want to settle out of court or if their offer is not enough to cover your loss, then you can proceed with a lawsuit against your neighbor — but in most states the fault will be distributed between the parties. If you fell due to your own negligence or a failure to follow safety precautions when using the ladder, your neighbor may not be at total fault for the loss. If your neighbor provided a faulty ladder that broke, then there may be a different outcome.