Day trips from London
 

Brighton and the Royal Pavilion would be near the top of my list. I'd also visit Canterbury.

Canterbury
 

I would. I'd also look at Warwick.

Here's a ringing endorsement for Winchester. The following is an excerpt from a trip report I wrote:

"Next morning we passed through Kingsgate, under tiny St Swithin’s Church, around the close and into Winchester Cathedral. This is a simply astounding place. Winchester was the ancient capital of England under the Saxon kings, and for a period under the Normans. Construction started in 1079 (the early Norman style was replaced with fine Gothic work in the 14th and 15th centuries). In addition to the architectural details you can examine archaeological finds from various epochs, see the mortuary chests containing the bones of early royalty, including King Canute, the 12th century Winchester Bible, a fine old library with two 17th century globes (one of which shows California as an island), a chapel in the south transept dedicated to Isaac Walton, author of “The Compleat Angler” where a corner of the stained glass window shows the master at rest on the banks of the Itchen, with the quote “Study to be quiet,” another showing him on the banks of the Dove with Charles Cotton, and his grave, where some anglers are said to touch fishing hooks to the stone in hopes of acquiring some ju-ju, the grave of Jane Austen, and another memorial which brought to mind present times, dedicated to one Edward Henry Le Marchant, who died at age 45 in 1899, “shot by a fanatic in Peshawur.” We spent over two hours in the cathedral.

Uptown we visited the Great Hall, with its supposed Round Table of King Arthur, site of the bloody assizes under Lord Jeffries (you might remember the scene from “Captain Blood”). Then downtown for a late and quick lunch, further past the statue of King Alfred, crossd the bridge over the river Itchen where we saw a brown trout finning in his lie, past a splendid half-timber building housing a restaurant called the Chesil Rectory, through the ruins of Wolvesey Castle, and out along the river to the water meadows, with Winchester College on our right. We saw an otter (much despised by Walton) swim to our side of the stream and disappear under the bank, which made my wife a bit uncomfortable.

Our final stop was Winchester College, founded in the 14th Century by Bishop William of Wykeham, where we caught the last tour of the day. Our guide was a charming woman whose son had been a day boy there. She brought us by the quarters where the scholars live, the classrooms where they and the commoners study, the chapel with its wooden fan vaults and Fromond’s Chantry with its beautiful cloisters where even today classes are held in summer. We saw the perfect servant, a college mascot, and the school’s motto “Manners Makyth Man.” . . . Our guide had tremendous verve -- she completed each stop by directing us on, then charging before us, her blue cape flowing behind. I was reminded of Maggie Smith’s performance in “Lettice and Lovage” which we saw in London long ago. Should you ever visit Winchester, do not miss a tour of the college. We returned to our hotel past the pale yellow house where Jane Austen died."

Another endorsement for Winchester.
We also enjoyed Canterbury, although we did arrive there with a car, not via train.
We loved Chester, too, but it also was with a car, on our way from Ruthin to Manchester, last day of an almost 3 week trip last June. I'd save it.

Winchester and Canterbury
 

>>1. Leeds Castle (this one is on my list for sure)<<

Not quite sure what you expect from Leeds Castle but me personally . . . I'd definitely put Hampton Court at number 1, and Winchester also well above Leeds C. on my wish list. Warwick, Arundel, Brighton etc would also top Leeds. Leeds is a 'pretty' castle, has lovely grounds, but there isn't much there there. It really is primarily a wedding/conference venue.

Have you considered Hever? Edenbridge is a 40 minute train ride from London Bridge and then a 2 or 3 mile taxi ride from there (or it is walkable)

Or Knole - about 1.5 miles from Sevenoaks which is a 35 minute train ride fro Charing Cross

Or IN London -- Eltham Palace which is a 25 minute local train ride from central London.

Hever Castle
 

Oh - I misunderstood that you had been to HCP before. So It can go farther down your list and fit in a re-visit if you can.

>>Leeds Castle and maybe Hever Castle are very weather dependent... I can book cheap train tickets now, but what if it will be pouring rain on that day?<<

London to Edenbridge is only about £13 no matter when you book even at the station as the train is pulling up.

Sevenoaks is about the same or even 10 or 20 pence less.

Absolutely no reason to pre-book either one

Prebooking Train tickets
 

Have you been to Hatfield House? If not, I would recommend it. Really easy to get to from London, 30mins from Kings Cross. The House is right opposite the station.

House | Hatfield House

Ham House might also be interesting, its close to Richmond which is a very nice area of London, if you like gardens, Kew Gardens is worth a visit

https://www.nationaltrust.org.uk/ham-house-and-garden

https://www.kew.org/

>>I know we pay extra for train tickets by doing that<<

That applies to longer distance train tix (like probably Bath on one of your previous visits. But those trip it Motingham (for Eltham), Sevenoaks, etc) are really commuter trains/distances not costly at all.

If you are going to the ROH right after visiting the National Portrait Gallery -- you might looking into booking the Portrait Restaurant's pre-theatre dinners. Terrific views, really good food and service, and quite close to Covent Garden.

Didn't see Odin's post -- Hatfield House and/or Ham House are quite good ideas

Also have you been to Kenwood House on Hampstead Heath? f it is a nice day you could walk up from Hampstead across Parliament Hill (Wonderful views across all of London), to Kenwood and then a late lunch or early dinner at the Spaniard's in walking back down into Hampstead.

(See - you have to add 10 days to your trip ;) )

Hatfield House, Ham House, Kenwood House
 

Probably 3/4 of my travel is solo and I try to have at least a few upscale to very upscale meals every trip. In London I 'eat posh' many evenings but some of the top tier places won't take a booking for one. The Portrait doesn't blink an eye -- I've eaten there probably 8 times or more over the years. Another place in the same area is L'Atelier de Joel Robuchon -- singles are usually put at the bar but that is great because it is an open kitchen and it is a great place to sit. And Outlaw's at the Capitol is wonderful for solo diners -- a VERY nice room. Also done single bookings at the Ivy -- but I seem to recall last time I checked the website only allowed for 2 or more. Probably a phone call would work though. The Wolseley always accepts singles.

The nice restaurants at the galleries and museums (the Portrait, the Rex Whistler @ Tate Britain, Level 9 @ Tate Modern, etc) will all take single bookings. And the nice venues in all the Department stores

(I had to lie my way into Gordon Ramsay Hospital Road -- could only book for 2 and when I arrived it was unfortunate "your friend had to return to the States unexpectedly. . . but sorry we'll have to charge you £200 for the no-show" :omg: . But in the end they didn't and it was never mentioned again.)

Great dinner suggestions
 

" Wolvesay Castle " I live not far from Winchester, and have always found this to be somewhat of a disappointment. You mention Arundal castle, and doubts over it becoming an amusement park - If you want a "pure" Castle, can I recommend Portchester?
It is less than an hour from Winchester - a direct train and about half a mile walk. It is a slab sided Norman Keep within the Roman walls. The audio guide is very quirky - the whole tour narrated by two ghosts, one English, one French (It sounds ludicrous, but it is very well done, and quite funny at times).

Portchester
 

I love Winchester, I used to work there on occasions and ran many of my cases at the Crown Court. I loved taking my lunch break touring the Great Hall and never tire of it, it's certanly a city I recommend as a day trip.

I don't think Arundel Castle has turned into a theme park of sorts. Sure, they've tried to make it more engaging and interesting for children but it's certainly not in the same vein as Warwick Castle.

I also completely agree with the suggestion about Portchester Castle. Fortunately for me I only live 15 minutes away and have probably overdone it with ny visits however it can easily be combined with a visit to Portsmouth in general, mainly a trip to the excellent Historic Dockyard.