I lived there for several years before moving back home. if you're a culture and food person, here's what i'd do
Museums - there are way too many to cover in four days. the must sees for me are the British Museum (where you could easily spend two or three days) and the natural history museum (as much for the architecture as for what's inside). There's lots more, of course, so pick and choose depending on what you like.
Music - if you like the opera, there's the national opera. the camden area has a very nice independent live music scene
Theatre and musicals - The west end is full of it. check timeout london (magazine or online) to see what's on. i didnt see as much as i should have, but i've seen les mis, and it was good. there's a lot of really famous ones like wicked, billy elliot, etc.
Being indian, and a complete foodie myself, I can offer up some recommendations. There's a restaurant called Tayyab's in Whitechapel that is bloody brilliant for what most people in the UK think of as indian food (which is essentially Punjabi food). There is also a hidden gem called Ragam on Cleaveland Road just next to the BT tower that serves south indian cuisine (ask for a masala dosa, and tell them to make it crisp).
But of course, you don't want to restrict yourself to just Indian food. Save that for when you're here in india. london is a foodie's paradise. there's excellent ethiopian to be had on caledonian road, excellent middle eastern to be had around edgeware road, and so on.
Foodie heaven, to me, is the sunday afternoon upmarket at Brick Lane. Its essentially an alternative fashion clothes market, but it has a section devoted to food, and there's lovely food from around the world. the ethiopian and moroccan stalls are my favourite. And clothes shopping at the up market isnt that bad either, though its turning pricey.
Do make it a point to have 'high tea'. there is nothing more sinfully enjoyable than good british tea (which, of course, is either from assam or darjeeling, so indian teas, but made in the british way) with scones. Most high end hotels do it, but they charge a bombshell. my favourite place that specialises in high tea is bea's of bloomsbury. its a small delightful little shop, and they do all sorts of wonderful deserts as well.
Also, the weekend markets are great places for food. my favourite market for food was the borough market - best to go early on a saturday morning. the cheeses, the chocolates, the cheesecakes... lovely.
There's two kinds of shopping in london, really. there's the high street shopping, which would include oxford street, new bond street, the westfield mall, etc. then there's the weekend market shopping, which is by far more enjoyable. my favourite was the sunday camden market, but that's because i lived there. brick lane is good on sundays, then there's the portobello road market.
Have fun! london's a lovely place, if you accept it for what it is!