It has taken me a long time, but I was finally able to get Last Comments working with private blogs! The version of this gadget for private blogs is Last Comments 1.3. You can add it to your blog by clicking on the link at the right (or click here). You can also host it from your own website by downloading it directly from my project repository and storing it on your own server (link to the gadget's XML file and link to the required image that must be stored with it). If you host it on your own server, you will need to change the required image's source location in the XML file (search for the <img> tag with an id of "LCAuthSubPixel") to the new location on your server.
This is the initial release, so there may still be some problems with it. I have not been able to verify that it works for visitors who have permission to read private blogs, but I have confirmed that it works for blog owners. Version 1.3 resembles version 1.2 in appearance outside of the login process. Please try it out if you are interested and let me know how it goes.