Your puppy is playing with you, stop this particular game.
Do not give her a command that you cannot enforce immediately. For now, she has not earned the right to run all over the house. She does not understand LEAVE IT. Usually if the dog starts running from you, if you run in the other direction, the dog will turn and chase, kool use the handle and catch her. But do not make a game of it.
But the first thing is, she has to be a whole lot better at LEAVE IT, and maybe an emergency DOWN, before you let her off lead. What happens if she gets a hold of a bottle of IB profin? If you have a solid DOWN, you can then walk up to her and collect the bottle and then release her -- normally in this situation, I would say, What a good Girl, lets get you a treat -- that is praising for the down and giving up the bottle. Then you walk with her to the fridge and give a piece of cheese or other high value treat.
But for now, when she is out, she has a lead on, until she learns to leave the cat alone, will do a LEAVE IT. Will do a DOWN. Then you can give her some time.
I do NOT like the idea of the crate for punishment. Do you have an x-pen? When you cannot supervise, use an x-pen right in the front room where she can play and be a part of the house and family, but you do not have to have your eyes glued on her. So when you let her out of her special play area. If you are done interacting, put her in her special place.
I do not like all this talk about punishment. First a dog has to know the command, then they have to know the command when you are sitting, in different places, in different levels of distractions. Even then, punishment makes little sense. A quick well-timed correction (usually a voice EH!) is enough or redirect, My Kitty, here is your bone -- that may work. Or you then put her back on lead.
A correction or punishment does nothing after the fact. If it takes a minute or more to round up the dog, the dog has NO CLUE why you are angry. He will NOT associate chasing the cat to being put in his crate. He might think cool, mom is playing with me, and after we play we have a nap in the crate. 3 minutes after -- no, he is just having fun. He may be avoiding being put back in the crate. He has NO skills to realize that he can avoid crate time by leaving the cat alone.
When the pup is ON LEAD, and he starts to go for the Kitty, THEN you correct with an EH! at first redirect him, and then give him a chance to redirect himself after the EH! and if he does praise, if he does not redirect.
Put him in the crate or in the x-pen when you are calm. It his his good place. Not a punishment place.
Bear Cub, Hepsi-Pepsi
Cujo2, Karma Chameleon
Ramona the Pest, Kojak -- who loves you baby?
Tiny Tinnie, Susie's Uzzi, Kaiah -- The Baby Monster.