I'd say Old Bailey yes, St Paul's yes, Temple of Mithras for teenage boys - no, they'd find it boring, unless they're real history buffs, it's ok, rest of them definitely yes.
But you don't want to drag them around loads of history places which they might find boring. Same with art galleries, if they like art then do that. Imax is always a good choice. And unless they're really into cats then no to the cat cafes, I've been to 3 so far, Lady Dinah's is the best one.
Hackney is good, there's a really nice go karting place near Mile End, Victoria Park is amazing with lots to do, you could do some sort of tour through Shoreditch, stop at Bar Kick (table football), go to Hoxton, Dalston, do a Brick Lane walk when they have the market and for a curry, down to Whitechapel and their art gallery, I think they even have a cartoon gallery there, maybe Photographers Gallery too. Greenwich is pretty amazing if they like boats and ships and the market is nice, you can go to the Observatory and then to Blackheath. The foot tunnel and DLR might be nice.