The text you highlighted contains two very common idioms:
That [x] of his.
When people say "that [x] of his/yours", it is usually said disparagingly. For example:
That dog of yours kept me awake all night with its barking.
Instead of saying "your dog", the inference is that the dog isn't even worth naming or referring to properly, hence "that dog"; yet the speaker also wants to apportion blame by saying that the dog is your responsibility.
"Keeping quiet" literally means to maintain silence. Keeping someone else quiet means taking action to ensure that the other person is silent, perhaps by keeping them occupied with something as a distraction, eg:
He kept his dog quiet with a bone.
However, "keeping someone quiet" can also mean to stop them from airing their thoughts or opinions, rather than literally preventing them from speaking.
"Keep quiet about..." can also idiomatically mean that you keep something hidden, that you do not talk about it yourself.
Without context, it is unclear to me whether your example means Dumbledore didn't speak about his sister for a long time, or if he stopped her from speaking (or from having her opinions heard) for a time.