Hi James,
This issue comes up every so often, and I'm not sure there's a one-shop stop answer for all of them, as situations seem to differ quite a bit. The most common solution seems to be the quality of the batteries used; sometimes just switching to a different brand is enough to fix the problem. Otherwise there are a variety of things to try and things to check, such as radio interference from other devices, dirt in the optical sensors, corrosion on the battery terminals and in your case there's always the possibility that a component has come loose inside following one of your drops. Here are a couple of answers you can look through and see if anything helps.
[post|748828|SOLVED: Logitech Pop Mouse Battery Drain - Logitech Mouse - iFixit|new_window=true]
[post|752365|SOLVED: Battery exhausting very frequently - Mouse - iFixit|new_window=true]
Read through and give some of the suggestions a try, then let us know what happens. If it's still not working, we'll see if we can figure out your next steps for getting your mouse fixed.