First, thanks for a great plug-in. I have your login form in my sidebar and it is now much more professional-looking for users, rather than having to leave the page to login. I have made a couple of small adjustments to your styling and the focus behaviour, which I'll share in another thread.
Second, my current question... I have an icon in my header that used to direct users to the wordpress login page. It is not really needed anymore because of your widget, but it is up at the top in an easily reached position, so I would like to keep it if possible and get it to work with your login form. Otherwise, users on small screens have to scroll down to login.
Is there an href or url address that could mimic the function of your login button, assuming the fields have been auto-filled correctly? (I think I can copy your code to check if the fields are already filled in, so this icon will only be visible when it is usable).
If this is not possible, that's fine. Option 2 would be for the icon to move focus to your widget so the user can then hit 'return'. (I turned off the line in your code that gives your widget automatic focus .)
I can post a link, if you need it, but the site is very much under construction.