The budget you will need for a trip to London can depend on a wide range of factors, including your travel style (luxury, budget, or mid-range), where you're staying, where you're eating, and what activities you plan to do. London can be quite an expensive city, but there are also many free or inexpensive things to do. Here's a rough idea of some costs as of September 2021:
Accommodation: Budget hotels or Bed & Breakfasts outside the city centre can cost from £60-£100 per night for a double room, while mid-range hotels might cost between £100-£200 per night. Luxury hotels can cost from £200 per night upwards.
Food and Drink: A meal at an inexpensive restaurant might cost around £15 per person, while a three-course meal at a mid-range restaurant might be around £30-£50 per person. Fast food like McDonald's is around £6 for a combo meal. A pint of beer is usually around £5-£6, and a cappuccino is around £3.
Transport: A single journey on the Tube costs £2.40 with an Oyster card or contactless payment card in Zone 1. A one-day travelcard for unlimited travel in Zones 1-4 costs £13.50.
Attractions: Many museums and galleries in London are free. Paid attractions vary in price - for example, as of my last update, a ticket for the Tower of London is £25 for an adult, and the London Eye is £30 for an adult.
Entertainment: Prices for theatre tickets can vary widely depending on the show and the seating, ranging from £20 to over £100. Movie tickets are usually around £10-£15.
So, for a mid-range trip, you might want to budget around £150-£200 per person per day, but it could be less if you're staying in budget accommodation, eating cheaply, and taking advantage of the many free attractions, or more if you plan to stay in luxury accommodation, eat out in high-end restaurants, and do a lot of paid attractions and shopping.