There are no night trains to the Lake District as the journey is too short and you would arrive in the middle of the night. Therefore you either need a one night stay in the Merseyside region or you need to stay another night in London and get the train north in the morning. What are you going to do with your luggage?
It’s confusing to call it the Liverpool ferry as it departs from the other side of the river, in Birkenhead, at least half an hour away by public transport. Stena’s website only tells you how to get there in a car which is a clue about how easy it is going to be as a foot passenger!
The two most obvious ways to spend the day are either in Liverpool (Beatles stuff, two very different cathedrals, Albert Dock, Tate Liverpool and the Maritime Museum or Chester (walk the walls, cathedral). I was in Chester a couple of weeks ago and Covid has hit the retail hard, with many vacant shops and restaurants. It had a depressing feel.
You need to check how far in advance you need to be at the port and allow plenty time to get there. Most routes involve a 10+ minute walk from the nearest transport stop, so you may wish to consider taking a taxi from Liverpool at that time of night.
Flying from London would be far less hassle!