Day 1: Parliament Square: Parliament, Westminster Abbey ( do the Vergers Tour or London Walks www.walks.com tour. Both good and informative.)
If no interest you can miss The Churchill War Rooms. If interested pre book tickets. Next to the War Rooms is the Horse Guard Parade. Check their website
https://householdcavalry.co.uk/museum/ if you desire to watch their changing of the guide or their ride up the Mall to Buckingham Palace.
From the War Rooms / Horse Guards its a pleasant stroll up Birdcage Walk in St James Park to Buckingham Palace. From the Palace walk through Green Park to Piccadilly. On Piccadilly you’ll find Fortum and Mason. Fun department store to meander about. From there head towards Leicester Square and there into Covent Garden. Lots of shopping and lunch spots. Dishoom is good. Now after lunch you can walk to the British Museum.
You could also try to have lunch in crypt of St Martins of the Field by the National Gallery.
Long day. Take your time. Use the tube.
Day 2: Tower of London. Go at opening. Head straight to the Crown Jewels. After seeing them see the rest of the Tower. Fascinating building especially when you look at the nearby Gherkin ( St Marys Axe). The old and new architecture of London. Near the
Gherkin is the Sky Garden ( free POV of London). You are also close to Leadenhall Market the last glass covered market in London. Not far away is Spitafields Market. Worth your time.
Definitely see theatre. The cast of Hamilton is terrific.
Day 3: Greenwich Observatory OR Hampton Court Palace. Both interesting.
Day 4: If its a Saturday: Maltby Market.
I’d do Brick Lane and see the Shoreditch Wall Murals.
Lastly, if adventurous you can ride horseback in Hyde Pack, use paddle boats in Regents park, stroll along the Thames path, wander along the canal path in Maide Vale up to Camden Lock and visit the vibrant Camden Market.
London is always changing yet always the same.
Great city to explore and appreciate. Have a great 4 days with your daughter.