There are a number of free roof top viewing areas that you might like to squeeze into your schedule (and who does not like "Free")
https://horizon22.co.uk/
https://skygarden.london/
https://www.8bishopsgate.com/lookout
Hey Hunter,
On a recent trip to the capital I had a delightful meal at https://megans.co.uk/ - it's mediterranean, so not exactly local, but delicious food nonetheless.
There's a few dotted around London, not exactly right next to any of the big places you mentioned but worth a diversion!