This is normal behavior, cats are natural hunters and your kitten is just "practicing". With my three cats, I simply give them a firm NO, and distact them using one of their toys. Also, kittens do grow out of the late night playtime, just as babies eventually sleep through the night. Cats are distracted with anything with "fur or feathers" on it. Or you can simply use a piece of yarn. I would recommend you get your cat neutered as well. In my personal experience (I have two males) it makes them less agressive, and they don't wander to far from home. It also keeps a lot of diseases away. Cats are highly intelligent animals and can be trained, if it is done with love and caring. Be firm, never yell. I have even taught one of my seven year olds and my eight month old to sit on command. Your pet will give you years and years of joy and laughter, just treat them right.
Best of luck.