Question: My bluetooth mouse batteries died, and when I replaced them, the mouse didn't reconnect. Now how can I get OS X to discover and connect my mouse without having a mouse to open the mouse dialogue?

Answer: 
If the batteries run out in my wireless mouse it never reconnects automatically and I have resorted to reconnecting a wired mouse to click on the bluetooth icon in the menu bar. Then I found that you can activate the bluetooth icon by pressing Control-F8 (the "Move focus to status menus" in the keyboard shortcuts) and using the cursor keys and the enter key to get to the connect menu item for the mouse.
Keep a cheap usb mouse and keyboard in the back of some closet (with the batteries removed) to quickly use in an emergency. I've successfully used the keyboard and mouse once each in the last 8 - 10 years but it's worth it.