How long ago did you buy the house? I’m not sure if you’d be able to hold the neighbor responsible for any pre-existing damage, only for any damage that followed a “Please trim your ivy at point X so it doesn’t get on my house” conversation.
I’d recommend politely asking your neighbor for access to his/her yard to cut the ivy. If he/she doesn’t particularly care about the ivy, you might ask if you could get rid of it completely. If the neighbor likes the ivy, though, I’m not sure what recourse you’d have other than asking him/her to keep it from spreading to your house.