It is generally accepted that Royal Caribbean are the best cruiseline for Children and activities. Stick to their Voyager class ships in the Med - Navigator or Voyager of the seas; or of course there is Independence out of Southampton but that is pretty sold out for this summer. You may be able to get a room when final payment time comes round in April as people find they cannot afford to pay the balance.
I agree do Royal Caribbean when my girls were mid teens they loved the teen kids club made loads of friends. Don't forget the Brilliance of the Seas also we got a fantastic deal sailing out of Southampton in May this year and these ships are still big entertainment centres just lack the ice skating and flowrider which are fun but the Brilliance has a cool slide and plenty for kids too.
We went on P and O's Oceana last year with our 13 and 9 year old boys. They have never joined in with kids clubs in the past, but this time we had to drag them away to join us for a meal. I would definitely recommend P and O.