This issue makes me laugh, you have skype and you attempt to login and even though your login credentials are perfect, it says "Sign-In Details not recognized". Even though you are on full unfiltered broadband, yet weirdly enough it might be working on another computer.
Now you probably Googled it and all the forums say "Download the Latest Version", I despise resolutions like that because even though sometimes it works, it doesn't solve the issue for those that do have the latest version, oh and since when do older versions just stop working?
Unless it's been announced that you have to upgrade, you don't have to do anything you don't want to.
This is a resolution with the latest or an older version.
Usually, this is caused by a corruption/break in one of three locations.
Now when you just uninstalled and re-installed you will probably find that Skype still has a problem. That is because Microsoft are lazy and leave crap behind and the Skype uninstaller leaves these folders on your hard drive.
So when you re-install, shock and behold it still uses the same corrupt files to run.
When you upgrade, you sometimes find that these folders are replaced or updated, thus removing the corruption. The upgrade doesn't spot and fix the corruption it just replaces files and folders that might or might not of been corrupt. This is why the upgrade sometimes fixes it, though depends on the update and what the update actually replaces.
Real Resolution (any version)
Uninstall Skype using Add/Remove Programs
Delete the C:\ProgramData\Skype folder
Delete the C:\users\%username%\appdata\local\Skype folder
Delete the C:\Users\%username%\AppData\Roaming\Skype folder
Reinstall Skype, ideally the latest version since you are doing an install anyway, but it doesn't matter.
Once installed, try to login again. Shock and Horror its working.
Other Stupid Resolutions
Another resolution (apparently) is install the latest version of IE. Just go away and stop blaming IE for everything, yes its full of bugs and yes everyone who has an IQ higher than 4 uses something else, but it isn't a thing to point the finger at when you quite simply don't know the real reason.
No. It worked before, it should still work. Software developers don't just go one day, hey shall we use different ports this time round.
No. It worked before, it should still work. Again Software developers don't just decide screw it lets completely revolutionize the way we connect through a proxy and not tell anyone about it.
You need to be shot. Unless there is genuinely a bug that Skype themselves acknowledge, there is no reason to downgrade.