Start at the Tower of London (Tower Hill Tube Station), cross over Tower Bridge going left at then end and find your way onto the Southbank. Take your time walking along looking at the river and absorbing the general londony-ness. Have a passing look at Shakespeare's Globe Theater, The Golden Hind, The HMS Belfast, take a stranger up the OXO towerlol, the winding alleys past the Clink prison and if you're feeling it go inside the huge looming Chimney that is the TATE to see some art. There are coffee shops and pubs along this route to keep your thirst at bay.
Continue on your path, under the bridge with the book shop underneath, past the BFI, the skateboarders and street performers. You should be able to make out the London Eye its huge, pre-book if you want to ride it. Could be worth it as a tourist only having one day to spare! Whether you go up or not is up to you, but you will be able to see the houses of parliament, Big Bens clock tower from your side of the river. Continue on, maybe stopping off in the Royal Festival Hall for some more coffee/art.
Now climb the steps to the bridge lined with the white spear things, slowly crossing and looking back at the path you just made. Walk up through and out of Embankment station, up the narrow road lined with shops to Charing Cross Station.
Once out and onto the main road cross over onto the other side. You now have two options. The First: Head right, five mins away is Trafalgar Square (home of the Free, National Portrait Gallery), leading onto Buckingham Palace (the latter is a pretty disappointing experience if you ask me unless you are aching to see some Queen's Guard) or the Second: Head left, five minutes away is Covent Garden! I suggest you see Trafalgar Square and head back on yourself to Covent Garden.
You can enjoy the street performers in the square and visit the many shops hidden inside the market area. Make your way to Covent Garden Station. From here it is a short walk to Leicester Square, which after passing through you will be standing smack bang in the middle of Picadilly Circus! Very London. You will probably be getting hungry now, after all you have been exploring for around 5 hours. YOU MUST FIND THIS PLACE TO EAT. http://pattyandbun.co.uk/contact/james-street/ it is around a 20 minute walk from your current location.
Hopefully after finding the best burger in the city, you can set an alarm and have a mini nap in a nearby square. Upon waking, jump on a Tube and make your way to Camden Town for an evening of live music and more food and drinks.