It is both correct, only the context is slightly different. In my opinion, sentence 1 is more about a personality "the busy lady", the one and only. In sentence 2 it is about a lady that is busy, with possibly more busy ladies. If that makes sense?

Hi Kate! This is an interesting one! Both are technically correct, but there is a big difference in style and inference between the two. "You are quite a busy lady." is a good, flat, respectful way of putting it - but see my note about "lady" below. "You are quite the busy lady." is a more archaic way of saying it, but today it has a bit of a playful, sarcastic or patronising edge to it. The inference with "You are quite the busy lady" is not that there might be no other ladies that are busy, but that being busy has become part of your fabric, and (because "lady" is mentioned) part of your womanhood. Because of this, many people might find this patronising or even offensive. It's a very subtle thing, but I would certainly avoid even using the word "lady" in this sentence unless I was talking to a close friend in a playful way. Professionally and normally I would say something like "You are quite busy, aren't you?" Hope that helps!

You are quite a busy lady.