It's a shame you missed 'Long Day's Journey Into Night' at the Apollo which ended in August. One of my favourite plays and one of the best American plays of the 20th Century.
Currently at this present moment there is alot of hype surrounding Stephen Fry's comeback in Shakespeare's Twelth Night which is obviously a must see if you are a Shakespeare fan but if not obviously you may want to look for something else. Shakespeare's Globe (not too far from Kevin Spacey's Old Vic) is another venue that does Shakespeare all year round:-
Stephen Fry in Twelfth Night, Globe Theatre, first review - Telegraph
Talking of the Old Vic if you're a fan of the genre classic American musical 'Kiss Me Kate' will be playing at the Old Vic in January. And there isn't a better time of year to be in the Old Vic which is a remarkably comfortable venue to get relief from the biting January cold. It briefly premiered in July at the Chichester Festival:-
http://www.guardian.co.uk/stage/2012...chester-review
If you're looking for a family show then you must see Philp Ridley's world Premiere of 'Feathers In The Snow' at the Southwark Playhouse. The Theatre is a cosy little venue under the arches of London bridge and right next to a London pub so be careful you don't walk past it. I'll be seeing the show myself in early December. Always good value tickets here are £16, £14 or £50 if you have a large family. This will be the last show here before the Theatre temporarily relocates to a new venue:-
Feathers in the Snow by Philip Ridley at Southwark Playhouse
This is a real family show and a magical story that should appeal to adults and children alike. If you're interested check the Southwark Playhouse website:-
Southwark Playhouse - Theatre + Bar
These are the shows that I know will be on at the time you will be staying here and it goes without saying that Shakespeare performances are on all year round at the Globe but please scour the London newspapers guides for the Culture sections before you come to make certain that you will find something for your taste.
Another reccomendation is 'The Mystery Of Charles Dickens' which is on at the Playhouse Theatre. Not sure how long this show is running but if you follow the links on WhatsOnStage you will be able to have a good browse and hopefully find something sccording to your taste. I haven't yet seen this production but have booked tickets for next week:-
The Mystery of Charles Dickens Reviews at Playhouse Theatre - London - Whatsonstage.com