The workaround is to click on the 'options' button beside where your username and password is entered and to there, click on the button labelled "force update". This of course will update your jar file, so if you are using mods, you'll likely have to reinstall Forge or whatever, but the log in error is fixed by this. You can from that point log in normally.
Er, your method is incorrect. Login Failed could be tons and tons of reasons.
Either Mojang login server"s" are down, network issues, firewall issues, Anti Virus issues and many many more reasons.
And this you posted does not techincally fix Login Issues, if you logged in once, you can play in offline mode. Also there is a section on the forums, as well mojang's Help section on their website to tell status of their servers.
From here you should ALWAYS check first if you have login issues. If the service for Login Servers is not green and healthy, that is most likely your problem, if its green and healthy for Minecraft Login, Multiplayer Sessions, Mojang Accounts "if merged minecraft.net to Mojang account or etc" you have a local issue.
Downloading and reinstalling minecraft wont fix "Login Issue".
Well, all I can say is that I was indeed having a "Login Issue", and the http://help.mojang.com/ offered absolutely zero help, that even after me spending a good half hour there searching and reading. When I went back to the login screen and tried what I did, I did indeed fix my own "login issue" using the method I posted.
No doubt login issues can be caused by many things and what was causing my issue and therefore the fix to it might not work for someone else whose problem is caused by something else. But the solution to mine was extremely simple and if even a few other people are able to benefit from my find, then my posting it was well worth it.
I think the main cause for this is you have to buy the game. If you just downloaded it from the site without buying then you cant actually "login". but buy buying then when you login it will just take you straight to the game.