How can I fix a brand new Magic Mouse that won't hold a charge?

Sorry, but there are no DIY repairs for the Apple Magic Mouse. Even Apple doesn't repair them.At this point, there is no way to know what may be wrong with it.

There’s only so much you can do without opening it—but there are a few things worth trying before writing it off.

• Start with the basics: use a known-good Lightning cable and a different USB power source (Mac, wall adapter, etc.) just to rule out a charging-path issue. When you plug the mouse in, leave it connected for at least 30–60 minutes even if the charge indicator doesn’t budge at first. After that, disconnect it, power it off and back on, and check the battery level again.
• If the mouse still instantly dies after unplugging, the internal Li-ion cell is almost certainly defective. As I mentioned, Apple doesn’t repair them, but the battery can technically be replaced—just know it’s a delicate job involving prying up the bottom shell, desoldering the old battery, and installing a compatible 3.7V Li-ion pack.

It's very likely that this "new" mouse was sitting for an extended period of time with little or no charge. Rechargeable Lithium-ion batteries, left in this condition, lose their ability to accept a charge, and this is what you are seeing.