We will be taking our 7th trip to the Caymans this fall for 2 weeks dividing our time between Grand Cayman (Turtle Nest Inn) and Little Cayman (Little Cayman Beach Resort) but last year we did a 2 week trip to GC and CB (Brac Reef Beach Resort).
In my opinion, LC has the best diving, CB next, and GC last; but there is very good diving to be found off of all 3 islands. When are you going? Little Cayman is more expensive than Cayman Brac and it books up fast so it might be hard to find accommodations at short notice unless there is a cancellation. Brac Reef Beach Resort is planning some remodeling in August through Oct/Nov so I don't know if they would have availability either.
I think you will really enjoy Cayman Brac (we love it!) but since you have 10 days I suggest spending 2 to 3 days on Grand Cayman and then going over to CB for a week and stay at the Brac Reef Beach Resort. But remember that when you are doing a 2-island trip you will have to factor the 18-24 hour "wait to fly" interval into your dive schedule.
There's lots of stuff to see and do topside on GC (not just the tourist attractions!) and there is excellent snorkeling and some good, inexpensive shore dives like turtle reef. Take a couple days to relax, look around on GC, check out your equipment at the Turtle Reef shore dive.
Cayman Brac has beautiful healthy reefs and walls and a sunken Russian Frigate that is an easy wreck dive. And if the weather is good and several divers are interested, for an extra $30 US dollars per person (or $35?), the BRBR dive op will take a trip over to Little Cayman one morning to do 2 dives on Bloody Bay Wall - so you can experience it and decide whether you would like to return to LC for a future trip. If you go to Little Cayman Beach Resort instead they will offer the reverse trip over to Cayman Brac to dive the frigate, if several divers are interested.
There are plenty of things to do on CB, but not really any tourist-type attractions. The BRBR provides free bicycles for the use of the guests, there is a nature/bird preserve, opportunities for hiking, climbing, visiting the caves, enjoying the pool or lying on the beach in a hammock. There is a free exercise room and a small spa/salon that is quite reasonably priced.
The resort offers other activities for guests, too. Like trivia games and movies and a great Friday night barbeque on the beach. Believe it or not, the Bingo game is a blast - everybody gets into it, including guests, staff and locals!
When we stay at BRBR we like to spend time enjoying the diving and the resort, but we also do some biking and walking and we rent a car for a day to explore the island. I would be perfectly happy spending all 10 days on CB if you are not interested in spending time on GC. Cayman Brac is a pretty, quiet little island with friendly people; I highly recommend it.
We've always seen some sharks in the Caymans, usually reef sharks and nurse sharks but they are not as common there as in the Turks and Caicos and the Bahamas. We see more turtles in the Caymans than in most other areas and usually lots of rays and morays and tarpons and colorful reef fish and corals and sponges.
The Caymans can be expensive (so are the T&C islands and the Bahamas!) so it is good to have a plan for your costs in advance. When we stay on GC we rent an apartment and a car and do a lot of our own cooking; but if you are only going to be there for a few days - that probably doesn't apply. Search this forum and ask for advice for places to stay and eat on GC.
I think that the BRBR is quite a bargain for the Caymans, their prices are in US dollars and they include accommodations, meals, valet diving, transfers, and hotel taxes/service charges. You just have to pay for your airfare, dive op tips, and bar or spa bill. So you will have paid most of the costs before you leave home. The food is buffet but it is excellent.
We also really like the Reef Divers dive op and their valet dive services (the same dive op serves both BRBR and LCBR). You won't have to handle your equipment all week (unless you want to) and they bring your gear to the back of the boat and help you into it - and back out of it later - so it is very easy diving. You can follow the divemaster in the water or go off on your own. There are lots of permanent moorings and most of the dives are wall dives with little current.
If you decide to stay on Grand Cayman, that's great there is plenty to do and see. And it may be more attractive to a young person than CB - depending on what you and your daughter like to do.
There are both good diving and poor diving options around GC and some great dive ops and others that are not as great, so do some research before you book. If you will be on GC for 10 days you should definitely rent an apartment or look for an all-inclusive IMO. You should also rent a car for at least part of the time. When you are pricing various packages and options make sure to clarify the currency. The $US is worth about .75 to .80 cents in the Caymans. Avoid the busier areas when cruise ships are in port. Here's a nice thread about free or inexpensive things to enjoy on GC:
http://www.scubaboard.com/forums/cayman-islands/437891-topside.html#post6532430
The Cayman islands are clean, you can feel confident drinking the water and eating the food, the communications systems are good, they have fine health facilities, the people are nice, and the crime rate is low. Have fun!