It just took me 10 minutes of trying, a search in blogger help & a change to my internet settings to finally get to post a comment on one of the usual blogs I follow. Same problem again, asks me to log in with email & password, takes me back to the comment to put in the codeword, takes me back to the same screen asking for email & password etc.
According to the blogger help I needed to go to Internet Options / Privacy / Advanced & Allow Third Party Cookies. Mine said they were allowed but I said override & clicked on the same again & I can now post, having just tried again on about 6 other blogs. Just in case this helps anyone else. They also said that comments were still working for those who havea pop up comments form rather than the embedded type so maybe that is worth a try too.
Hopefully happy commenting