I tried resetting the connections to both the mouse and the keyboard following the Logitech instructions, but the mouse connection was lost after five minutes or so. I found that if I just press the "connect" button on the mouse (without pressing the button on the receiver), it will re-connect the mouse for another 30-60 seconds.
The keyboard is still working fine. I've had the mouse and keyboard for three months with no previous problems.
In case this is related in any way: My computer recently crashed in a big way and I had to reformat the hard drive.