When do you think its a good age to start giving kids more responsibility?

My daughter is 5 yrs old and will be 6 in November. I think it is about time to start cleaning her own room and helping me out some around the house. Do you think she is too young? She also has been pampered for 5 yrs now and is very mouthy, she thinks she it the boss! I don't know how to get her out of this and I fear that Ive let it go on for too long NOW. When she wants to go swimming, we go when she wants to go to her nanny's she goes. Whatever this child wants she gets and if I ask her to help me out around the house she acts like Im killing her... I need help with this.