Well I had no difficulty at all in finding it - I got the hotel to send a taxi to meet me off the ferry!
The hotel's on Montenero, which is a few miles outside central Livorno, but means there is a fabulous view. (I imagine in summer it also means it's slightly cooler, with a breeze.) The hotel building is an old villa, with a terrace outside. Inside it's more just a normal hotel, with rooms on the ground floor and wide corridors, so it might be a good choice for people with mobility problems. Comfortable modern room, good shower, good breakfast, stupendously helpful concierge on front desk, who spent over an hour locating the Livorno residences of Keats, Shelley, Byron and Smollett for me.
I liked Livorno and I'd like to go back. If I do, I'll stay at La Vedetta again.