Casual and Fun
I was only in London for two days since I spent some time in Oxford beforehand, which I highly recommend if you are a little overwhelmed by London. I didn't do much per say, but I walked around all the sites and I had a super time just absorbing the sites for what they were. I did this both at night and during the day. I am going back to London based on my few days there if that says anything to the feel good vibes of the city.
Insider Tip:Go to Nando's and eat some spicy chicken! Random, but as an American studying in Europe, it is super hard to actually find spicy food. It's also fun and silly to try a local chain restaurant. Also, take the time to walk London. It is big, but it is nice to walk and see some things and save 4 pounds by not taking the Tube.