First, you don't want to train the dog to pee in her crate. A crate is a no-pee zone. Second, if she has enough space to pee in one corner and sleep in another- the crate is too big. What kind if puppy is it? If it's bigger than a toy dog, you'll likely be better off just housebreaking and teaching the pup to go outside.
And this age, that means VERY frequent potty breaks (every 30-60 minutes) where you take the puppy to the "right" spot and reward heavily when they go, then give them some free time and play when you go in. Inside, if you can not actively watch the puppy very closely, then the puppy should be in it's crate, which should be just big enough to stand up, sit down, and turn around. Take the puppy out immediately when it leaves it's crate. If your puppy pees in the house- it's because you were not watching it well enough- that's your fault, not the puppy's, the puppy doesn't know better. Keep a strict routine, and the puppy will learn quickly.
Keep in mind the max time they can be expected to hold their bladder is about an hour for month of age, at 8 weeks I'd take them out much more often, but even once they "get it", until their adults you have to keep them on a strict schedule and bring them out frequently, just gradually lengthening the intervals until they're full grown.
If this puppy is a toy dog, that may alter things a bit- toy dogs have tiny bladders and often can't hold it nearly as long as bigger dogs can.