If you decide to go to Brighton don't stay there for 4-5 days, I think it's too much, you better keep more days for London. London is huge and there's so much to see/do. If you will visit Hampton Court, then you can visit Kingston upon Thames in London. It's an beautiful Royal borough that lies on the River Thames. Richmond upon Thames is an amazing borough as well. If you want something different you can visit Camden. And in East London you can check out Shoreditch and Brick Lane.
Enjoy