Not sure if you're a pub person but I've organised a number around the City and you have quite a few not too far, A lot will be closed on weekends so this mainly works if you have a afternoon or some evening time to kill in the week.
I assumed a taxi or tube out to Brick Lane and starting off with the best Salt Beef Beigel in London to give a little lining to start ;)
If you didn't fancy that you could easily swap it round to a more circular walk from your hotel and save time, I've *'d the ones I think are don't miss.
It weedles you back to your hotel taking some of the sites of the city including being close to the Barbican if the architecture their interests.
Smithfield Meat Market and the Musem of London are also worth some time to wander round.
The Beigel Bake Brick Lane
https://bricklanebeigel.co.uk/
The Artilery Arms
https://www.artillery-arms.co.uk/
The Hand & Shears *
https://thehandandshears.com/
Closed Weekends
The Jerusalem Tavern
https://www.stpetersbrewery.co.uk/london-pub
Closed Weekends
Ye olde miter *
https://www.yeoldemitreholborn.co.uk/
Closed Weekends
The Seven Stars *
https://www.thesevenstars1602.co.uk/
Map Link -
https://www.google.com/maps/d/u/0/ed...V1&usp=sharing
If your free Sunday morning or an early riser, then Columbia Road Flower Market is an interesting way to spend some of Sunday morning, after you can head down via Brick Lane and catch the Salt Beef Beigel if you missed it earlier, have a wander in the Brick Lane Market which is mainly brikabrak now and possibly the Truman Brewery market, this opens up the Spitalfields Markets and another great but slightly down at heal pub 'The Pride of Spitalfields'
Something else to look at is the Wiltons Music Hall, it is as close to an old time music hall as you can get, it has never been renovated just preserved and is a wonderful old building, sadly I don't think you can go in at present if you are not attending a performance but if there is something on you like I'd recommend the experience or perhaps call and check
https://wiltons.org.uk/