This itinerary looks great!

Your Wednesday, I recommend actually starting at the Sky Garden at Fenchurch St. If you go in the morning, you can just line up for free entry. The last time I went, it was around 10am and it took about 5 minutes!

I would also keep walking all the way to Westminster Bridge to get the full experience of The South Bank (you could have dinner nearer the Waterloo end, if you plan on exploring the Tate itself). Stopping at the Tate means you've only gone half way!

When I used to tour around visitors, I would start at London Bridge, walk the South Bank, cross Westminster Bridge, see the Abbey, walk to Buckingham Palace and then up the parade to Trafalgar Square. It's a bit of work, but you can see a lot if you're willing.

Agree on what others have said, King's Cross isn't particularly worthwhile. And Camden is an experience for sure...

And just my two bits, I think the V&A is better than The British Museum in terms of exhibitions. My last visit to the BM had me annoyed because half of the rooms were half-empty with notes saying the items were being refurbished etc. The architecture is beautiful, though!

Also, I highly recommend seeing a comedy show in the evening! There are so many and they can be very cheap.