My first guess would have been United Utilities, but as you've already found out, they don't appear to own it.
So, I wonder if it's one of the other large landowners, like the Lowthers, or possibly the Dalemain estate.
Here's the contact details for Lowther Estates Lowther Estates
Also, the Dalemain Estate Dalemain
Failing that, it may be worthwhile giving the Lake District National Park a ring. LDNPA
Lorraine