I can agree with all of this. Borough market is nice.
The general South Bank walk is good (London Eye - Tate Modern - St Pauls - Globe Theatre)
Depending on your tolerance for shakespeare, a cheap standing ticket at the globe theatre can be fun… if the weather is good. Or there is also the open air theater in Regent’s Park (if the weather is good).
Since the weather is likely to be variable, the museums (and associated museum cafes!) are great for rainy days. The V&A is across from the Natural History museum and is also really nice.
Buckingham palace is dull and ugly… unless you can get inside. Windsor is better, but packed with tourists in summer.
Hampton Court Palace might be the best bet as it seems to be less well known.
Day trips: Oxford (if you like harry potter esque architecture - but lots of tourists). Brighton (on a sunny day) if you feel like something a little more modern.
Avoid anything calling itself “Ye olde Englishe…” or “the world’s best fish and chips”.
I haven’t been, but I guess The Shard viewpoint is the new thing?
I dunno if it’ll match your schedule, but for a different experience: http://www.thelunacinema.com/crystal-palace-park/4592086242