I Googled for an answer and came up with numerous posts about an issue with Internet Explorer 8. I decided to just put up with the issue and wait for Microsoft to bring out a patch.
However, I recently changed back to a standard sized font and discovered that the gadgets suddenly started working again. I had set a custom font size at 115%, which is 110 DPI — slightly larger than standard 96 DPI, but less than the large size of 120 DPI. On reverting back to the standard 96 DPI, my sidebar gadgets — including the clock, CPU monitor, calendar and weather — have worked perfectly.
To access this setting, right click on the desktop -> click on personalise -> then change font size (DPI).