Using the latest thunder bird and installed your extension but there is no button, nothing. I enabled "sidebar" in the view menu but to do ist ls is blank when it pops up.
So the Todoist sidebar does appear, but the content is blank? Please try clearing the cache for Thunderbird (or Firefox, same thing) and see if that resolves it.
Do you have any other extensions installed for Thunderbird, and is it accepting all third party cookies?