I'd recommend spending most of your time at the away from the West End. See some shows at the non-profits. The National Theatre, Old Vic, Young Vic, Almeida, Royal Court, Globe, Barbican, Donmar Warehouse, Menier Chocolate Factory, etc. That's where the exciting theatre is happening. Even more so than New York, non-profit theatre is really the heart and soul of London theatre. West End has some good offerings for sure, but you'd be doing yourself a great disservice to fill all your show slots with commercial theatre.
And I'm not just saying that because I'm a big fan of non-profit theatre (though I am). I think the average out-of-town theatre fan can fill an NYC theatre trip with Broadway shows and they still may not see everything good on Broadway. They may miss some great things Off-Broadway, but usually nothing that's an absolute must-see for someone with limited time (and this coming from someone who sees mostly Off-Broadway shows).
Commercial West End theatre is not the same IMO. It's really not where the hot, relevant, must-see theatre is happening.