How can I resize the screen to use it ? (attached picture)

I am not aware of a way to do what you want. For many things you could simply drag windows further right but things like the Apple menu will be stuck.

Apart from the obvious solution of paying to get your screen repaired an alternative is to plug in an external display and run the laptop in either 'mirroring' or 'clam shell' modes. With mirroring the external and internal screens will display the same image, with clam shell mode you close the laptop and this disables the internal screen - you obviously in clam shell mode then need to use an external keyboard and mouse.

In the Displays Pref Pane choose Scaled & see if a different Rez helps.