Please try to click on the TD icon in the notifications area (system tray) in the bottom-right corner and use the Logout option. It should redirect you to the login page.
If this won't help, try to install this version: https://d2dq6e731uoz0t.cloudfront.net/46df05282d620e14d9ddd8a60d813dad/as/Todoist_for_Windows.RC.exe and let me know if the issue persists.
Also, please make sure that you're online when you try to access the app and that there are no firewalls blocking the connection of the app to the internet.
It doesn't work. Logging out does nothing on either version.
It may help that I am under a corporate proxy server, but it its correctly configured on IE and the web client works without a problem.
We are aware of the proxy issue. We hope to have it fixed as soon as possible.