Take a very thin piece of hard plastic and run it around the button seam to dislodge any crud and/or obstacles that may be impairing function. If this works then it would be worth addressing the battery issue. Let me know if the click wheel is restored and we'll proceed to the battery. Here's the battery: iPod 3G Replacement Battery
Replacement 850 mAh battery for iPod 3G. This replacement battery has 30% higher maximum capacity than the original battery. Includes free set of iPod opening tools. — 19 available at 13.95 each.