Hi, I’m from the North East and have lived in London over 10 years... let me try to think based on when my little brother used to come down.
South Kensington is where the science and natural history museums are (both free) get there early as they get very busy. Personally I think the v&a is a more adult museum but if your kids are into design it’s good.
Covent Garden and Leicester Sq are popular, tons of restaurants around there. Kids seem to like MeatLiqor (burgers etc), Rainforrest cafe, there’s a Forrest Gump themed restaurant in that area too. Soho is really good for food, or maybe China Town would be fun for them.
In terms of shows, you can get last minute tickets in Leicester Square or there are some good apps - maybe someone can help with the name of them. Matilda, Lion King, Charlie and choc factory all really good.
Tower of London is also an option - expensive but lots inside to see. I think the museum of London is good too, it’s opposite st Paul’s.
Finally, nice to walk along the river from London Bridge to Waterloo... you go past the globe theatre, the Tate, aquarium etc... lots of pubs to stop off in.
Let me know if you want any more recommendations aware I’ve written a total essay! 😂