Let's not neglect the obvious. First make sure that your mouse has fresh batteries in it and that the batteries are in the correct orientation. Slide the switch on the bottom of the mouse to see if the green light comes on. If it does, the electrical contacts are good and you're done.

If no joy, examine all four battery contacts inside the Magic Mouse's battery compartment. Scrape off any brittle white material or corrosion until the metal contacts are mostly shiny where they contact the batteries.

Reinsert the batteries to see if that fixed the problem. If that doesn't work, carefully check the negative battery posts.

The negative end of the AA batteries are supposed to press firmly against shiny, spring-loaded metal posts. If there's too much dried battery leakage gumming up the posts, they might be partially or fully stuck under compression. If so, weak or absent spring action will cause the battery contacts to fail.

To deal with this, carefully scrape all of the visible dried battery residue off of the posts. Press the posts and release them several times, trying to free them up until they pop up vigorously.

If that doesn't work, try shooting a very small amount of WD-40 at the base of the battery posts to see if that frees them up.