hi!
i worked in Monaco for 4 months and I live quite close so I maybe could help her too.
children could do most of possible sport, they have a lot of place. swimming pool close to the harbor, rugby, soccer, rowling, I know that the prince goes to an indoor place which is public (not when he goes) where you can practise a lot of different sports.
then you have 3 museum, you have a library...
then you have a lot of tradeshow or events during the year. Formula one race (a huge and noisy event) a horse contest...
and finally if you don't find something you want it's easy to take the train to nice or other close cities (even in Italy.
i hope i helped you a bit.