London in March with boys ages 11 and 9

Leeds and Warwick are both good castles for boys. Also Hever. Dover is good as it has not been "civilised" into a house by later generations.
Please note, that is Leeds in Kent, not the much larger city in Yorkshire.
There is plenty to do in London for 6 days. If you had longer I would recommend York. That has real city walls, and city gates, that you can walk along the top of and imagine shooting down arrows from. Plus the Railway Museum, great for boys of 9 to 90. I enjoyed York at that age (9).
Picture here: http://romancebandits.com/wp-content/uploads/2013/12/York_MicklegateB_Dec2013.jpg

Please don't abbreviate English places, as this causes confusion. Do you mean Leeds, a city 175 miles north of London or Leeds Castle in Kent? Leeds in Yorkshire - you don't have time for this and it wouldn't be my choice of time out from London. Leeds Castle in Kent - interesting, but Bodiam Castle, owned by the National Trust is a better example (it's a ruin) and has a great heritage steam railway nearby.

Dover Castle is a full day trip from London, so you may feel it takes up too much of your limited time. Warwick Castle is owned by the same group as Madam Tussauds waxworks, which is enough to put me off.

You could stay somewhere in Kent for a couple of days and pick up Dover, Hever and Leeds Castles, but you would need to hire a car to get around. Alternatively, just stay in London and do day trips. Windsor Castle is another possibility.

London Transport Museum in Covent Garden is popular with boys. So is the Operating Theatre Museum for ghoulish reasons. Take the river boat down to the Royal Observatory in Greenwich.

There is plenty to keep you occupied in London for the time that you are there. Many of the museums are free to enter.

Do they like Harry Potter? If so, be sure to take the Warner Brothers Studio Tour. Our whole family loved it. You must buy tickets online in advance of the tour.

Would your boys read guide books? Once on a month long trip to Finland and Paris, my 9 year old son got a little bored on a long ferry ride so I gave him the guide book on Paris. I told him to plan our Paris segment which he did. I told him the things I wanted to see. He wrote out an entire itinerary based on when things were open. When we arrived in Paris he was very interested in doing all the activities that he had planned. Even the things I wanted to do....

The Tower of London is full of armor, weaponry, etc...my son loved it when he was your boys' age.
You could head out to Oxford for the day and visit several sites where Harry Potter was filmed...Christchurch College, the Bodleian Library, etc.

"The Tower of London is full of armor, weaponry, etc...my son loved it when he was your boys' age."

+1.......And is officially a "castle".

Another vote for the Tower of London. I don't have boys, and never was one, but this place is much cooler than I thought it would be. Beefeaters, weapons, armor, ravens, beheadings... sounds like a great place for kids. Of all ages.

Hampton Court. You'll take a train to get there. There's history and beautiful gardens although might be a bit chilly in March. They were virtually deserted last November when I visited.

Another vote for the Tower of London. Crown Jewels, ravens, armory, etc. If no one has vertigo I'm certain the Tower Bridge Experience would interest them. Mudchute Farm. Petrie Museum. Household Calvary Museum. Bike ride in Hyde park. Whitechapel Bell Foundry ( check their website for open hours). Experiencing the hustle and bustle of the London Underground. A cruise on the Thames. The view from the Skygarden. The Horse Guards parade grounds.

Take them shopping at the Hoxton Street Monster Supply Store. You'll find it amusing as well. If you happen to be near St James Park at 2:30 there's a nice free diversion and that's watching the pelicans being fed. Happens next to Duck Island Cottage. The Royal Air Force Museum in Colindale. Can get there by tube. Natural History Museum. And for the best POV of London sit upstairs in the front seats of a double decker bus. LOADS of things to occupy the minds of adults and children alike in London. Will be chilly in March so remember, layers.