If you have an interest in seeing the exhibition at the Queen's Gallery you can get a combo ticket for Buckingham Palace and the QG. The exhibit this summer is one I've been looking forward to and have scheduled time in London based on viewing it! It's a celebration of the 500th anniversary of the death of Leonardo da Vinci with a show of his work from the Royal Collection.
Only book the Queens Gallery if this exhibition sounds interesting to you, otherwise just do the State Rooms at Buckingham Palace tour. There is a nice audio tour that goes along with the State Rooms.
I enjoy seeing the Royal Mews as well which can be bought as a combo ticket. To me the State Rooms takes about 2.5 hours excluding time thru security and in line (even with a timed ticket). The Mews takes an hour or so (be sure to go to the bathroom there - the tiles in the Ladies' Room are beautiful!) If the Queens Gallery exhibit appeals, book the Day Out with has all 3 venues - all accessed along Buckingham Palace Road on the left side of the Palace as you face the front.
I have an interest in WWII and have easily spent 3 hours in the War Rooms and that is not even covering everything in the exhibit on Churchill....just a quick run thru there and then listening to the audio all the way thru the War Rooms section.
I'd not do both of these in the same day since you've got 8 days.