You know that your cat is independent and self-reliant.  However, her happiness and health depend on you, her owner.  Brushing can be an enjoyable experience for both of you. The brushing sessions should be short until your cat learns to like it.   When cats groom themselves, they ingest a lot of hair which can lead to hairballs.  Cats with long hair need to be brushed daily to prevent matting.  Short-haired cats should be brushed two times a week.

Your cat's nails should be clipped about every two weeks.  If she spends time outside, she may wear down her nails more than an indoor cat and thus need clipping less often.   Be sure to touch and massage your cat's paws without the clippers in your hands.  This will accustom her to your handling her paws and the clipping should be easier.  

Be sure that you clean her litter box often.