Please carefully read  and follow the instructions in Troubleshooting wireless mouse and keyboard issues

They may be new batteries but if the old ones were corroded you may have to clean the compartment out.

Here are a few notes I've put together on batteries.

Whether you use regular or rechargeable batteries you should buy the best. Discounted or unbranded batteries are more likely to give a poor performance and may be of an irregular size, often leading to connection problems. Apple’s own brand, Eneloop and Energiser and Duracell appear to be the best. Rechargeable batteries for the standard Magic mouse are unlikely to exceed eighteen days before needing a recharge although the mouse itself can be a limiting factor. I have seen reports that some Logitech mice vastly exceed this. Apple may suggest more in their sales blurb but this is a realistic expectation. The benefit (if you are organised) is that you can always have charged batteries available.

Regular batteries will give you between 25 and 35 days before they need to be replaced.

Note. Never mix old and new batteries or rechargeable and regular. The notes above are based on my personal experience, not shutting down my machine and using it for around four hours per day. Note. Monitors do not always faithfully report battery strength, particularly the rechargeable ones.

When checking your mouse or keyboard batteries ... Turn off the device. Remove the batteries, Clean the terminals both inside the battery compartment and the batteries themselves. Replace or renew both batteries. Turn on the device. You should get a blinking green light if the device is paired. Click the mouse to activate it. The green light becomes steady.