Windows won't close and have broken locks -- how to fix?
I won't address the "broken locks" part of the question because the answer is self-evident.

You have what is commonly called a tilt window that uses spiral balance mechanisms to keep the window up when you open it. Common reasons for these windows to become difficult to fully close are worn or damaged spiral balances or worn or damaged balance shoes.

The balance shoe is a part that rides up and down in a frame channel on the side of the sash, the window sash connects to it at the bottom corners. Sometimes you can just lubricate the channel that the balance shoe slides up and down in, to make it easier to open and close the windows.

Check out this video to familiarize yourself with how and where the balance shoe operates. There are many more online informational resources to find by searching spiral sash balance.