As I understand the law (nb I am not a lawyer), no-one is responsible for fencing their own property unless a specific duty is included in the original deeds.
You can't force someone to either erect or repair a fence between their property and yours.
Equally they can't stop you putting up your own fence - as long as it is on your land and observes any local bylaws or planning constraints.
If the ownership of the original fence is disputed/unclear then the simplest option is to erect a new fence along your side of the old one - you lose a few inches of land, but that is probably worth it to avoid continued aggravation.
The T shaped marks on a boundary line on plans are commonly supposed to indicate ownership/responsibility for a fence, but unless there is a specific statement in the deeds this is debatable at best.