Hi,
I have been to alot of the islands in Thailand, also to Bangkok several times.
I would suggest 3/4 nights in Bangkok,
7 nights in chiang mai, and then a beach for 7 nights.
Krabi is very nice, ao nang area has some lovely islands you can go to,and beautiful scenery. Your daughter would enjoy looking around all the little stalls and shops in the evenings, and there are lots of restaurants to choose from.
It would definaitly be cheaper to diy, look on asiarooms for hotels. This site sometimes gets bad reviews, but i have booked on here lots of times,never had any problems at all,just got what i wanted, great hotels at good prices.
In Bangkok,the marriott and spa is a excellant hotel, right on the river, lovely rooms,and very spacious pooland garden area. superb buffet breakfasts as well.
Have fun
suzi