There are several venues to see great artwork for free at night in London. Here are a few options:

The Whitechapel Gallery is a public art gallery located in the East End of London. It is open until 9pm on Thursdays.
The Barbican Art Gallery presents major exhibitions by leading international figures, from architects to Turner-prize-winning artists. It is open until 8pm on Thursdays and Fridays.
The National Portrait Gallery is an art gallery that offers a collection of portraits of historically important and famous British people. It is open until 9pm on Fridays and Saturdays.
Victoria and Albert Museum is the world’s largest museum of applied arts, decorative arts and design, housing a permanent collection of over 2.27 million objects. It is open until 10pm on Fridays.
The National gallery houses a collection of over 2,300 paintings dating from the mid-13th century to 1900. It is open until 9pm on Fridays.
Some well-known public art areas:
Brick Lane offers some of the best street art in London and there’s everything there from murals of colourful flowers to giant elephants.
Dulwich Outdoor Gallery is a collection of street art in south London, with works based on traditional paintings in Dulwich Picture Gallery.
Leake Street has an underground tunnel that is decorated with amazing graffiti, initially created during the Cans Festival organised by Banksy on 3–5 May 2008.
Shoreditch generally runs from the area of Brick Lane up towards Old Street. There are many places to see street art along the way.
Leicester Square has a statue trail called ‘Scenes in the Square’, which celebrates a century of cinema.
Please note that exhibition timings and availability are subject to change, so it’s always a good idea to check the respective websites for up-to-date information.