It depends on the sort of shopping you are talking about - expensive items? Obviously, that would push the cost up (and perhaps you or your partner have already been talking about things you would like to buy so it is creating the idea that this might, to some degree, be a shopping trip).
If not, I would think £600 is more than enough unless, as others have said, you are planning on eating at the finest restaurants. Even then, you'd be hard pushed to spend that much.
As has been said, there are lots of interesting free things to do when you come to London - the British Museum, The Sky Garden (it is free but you have to book on line and queue for a little while), The Science Museum, The Tate and Tate Modern, other art galleries, etc, etc. A hop on, hop off trip down the River Thames to Greenwich is very affordable too - I believe about £15 each or possibly less. Just walking around is free - there's the South Bank, the Millennium bridge, leading to St Paul's Cathedral (which, if you want to go in is not free), Trafalgar Square, Covent Garden (with all the free entertainers and the interesting markets). There really is no need to spend a fortune - and perhaps unwise if purchases have to be made on credit. If you don't want to walk, the tube is relatively inexpensive.
As someone else said, do a bit of research before you go - eg for reasonably priced restaurants with good reviews on Trip Advisor. Have some idea of what you want to do or you will just be scurrying around with no plan as to where you want to go and what you want to do. It sounds like you only have one free day anyway as you have planned an activity on the other day, so you will have limited time.