I’m a retired English teacher (after 38 years) who has visited England several times; there are a wealth of literary associated sites, so enjoy planning and deciding which sites are most important to you and your interests.
When you say, “ten days total,” does that include your arrival and departure days, or will you have ten full days for touring? If it’s the first, then you would have about 8 ½ days for actually touring sites.
On your arrival day, it’s best to do outside walking tours (either self-guided or a London Walks tour).
Depending on where you stay, traveling to and from the Lake District takes 3 ½ to 5 hours by train and bus, so that will take up a good part of two separate days. If you have 10 full days and really want to visit the Lake District, it could be worth it, but if you only have 8 ½ travel days, you may wish to concentrate your time in other areas. There are plenty of places to hike and visit small villages that are closer to London.
The Lake District is beautiful, with lots of literary associations, so it’s hard to pass up. But think about the amount of time it takes to get there.
You could look at flying into London and then back from Manchester Airport, which is closer to the Lake District (and only a 20 minute train ride from Manchester Piccadilly. There are several hotels a short walk from the train station. I stayed at Motel One Manchester Piccadilly.
Here are some other ideas for literary themed visits:
Oxford has many literary associations, including Lewis Carroll, John le Carre, and Colin Dexter (Inspector Morse) in addition to those mentioned in earlier responses. Le Carre attended Lincoln College as did his character, George Smiley.
You also have the Bodleian Library and Blackwell’s bookstore (which is a delight).
There are nice walks along the rivers and bike trips as well.
You can book a short river trip, including an Alice in Wonderland trip, or take a boat to nearby Abingdon and walk or take the bus back.
https://salterssteamers.co.uk/short-trips/
https://www.localwalks.co.uk/local-walks-blog/oxfordshire-walks-abingdon
https://www.experienceoxfordshire.org/places-to-go/abingdon/
Oxford is a reasonable day trip from London or Bath, or you could stay a few nights and explore the Cotswolds and Stratford-upon-Avon from there.
You can get a taste of the Cotswolds in a day; here is a tour that’s been recommended by several:
https://www.gocotswolds.co.uk/product/cotswolds-in-a-day/
For more ideas, look here:
https://community.ricksteves.com/travel-forum/england/5-nights-in-cotswolds
In the Cotswolds, you can hike part of the "Cotswold Way," a long distance route. Just pick two villages on the route that are connected by bus, then take the bus out and hike back.
I have a few more possibilities, which I’ll post in a second post.
All the best,
Raymond