Question question on final APT. move out bill, plesae help!

My APT. contract is for one year, I bought a house last month, so
there is about two month gap, on the contract, it says if I break the
lease I will pay 85% of one month of my rent + any days before
next tenant move in,

I found a neiborgh she moved in my apt. afetr I moved out, so there
is one week gap, that is the manager required me to turned in the key
and moved out, so they have one week to clean the APT. to be ready for the
new tenant.

My thought is, there is no gap between my move out and the new tenant moved in actually,
she planned to charge me those 7 days afetr I turned in the key when they  did the cleaning.
plus 85% of one month rent, I am thinking do I need to pay it?

If I don't pay pay what they can do to me, they have all my information including bank account etc.
I knew they can hire company to call me and sent information to my credit report, what else? 

I just feel it is not fair, I already found the next tenant for them, and after I turned in the key
they did the cleaning why do I still need to pay? anyone has similar experiences? Thank you.

That is a chase property. She is manager hire by chase.

Thank you!