After you have cleaned it all (as the other comments explained already) and put a new battery in there, it is possible that the electric contacts are still not clean enough (especially the left one with that deep rust) and it doesn’t work with the new battery.

Then you have to go at the contacts with a small file or a small piece of sanding paper and scratch them very good, so that they get really very shiny right there in the middle, where it counts. Then it will work again.