It's wonderful to see a post from someone who recognises that 10-year-olds should be experiencig live theatre!
The Lion King is incredibly popular (with ticket prices to match!); it's definitely suitable for ALL ages, so it should certainly be on your list of 'possibles'. If I was taking a 10-year-old to a stage show in London, however, my first choice would have to be 'Stomp':
http://www.thisistheatre.com/londonshows/stomp.html
Many performance of big London shows (especially Lion King) are sold out months (or even years) in advance, so I certainly wouldn't recommend waiting until the day of the performance. If, however, you are looking for same-day tickets, there are only two places which you should really consider. The first is the box office of the theatre you wish to visit. The second is the ticket office in Leicester Square which is jointly run by the main London theatres. (There are hundreds of places in London which claim to offer cheap tickets for the big shows but you should generally avoid them).
Chris