Knappar funkar ej

Random Solutions and Workarounds Culled from the Comments (and Elsewhere):

A double-tap of the Windows (or Super) key.
A double-tap of specifically the right Windows key.
Restart the effected application (or Explorer if effected)
Reboot (last resort).
“Detect and Repair” from the Help menu in Outlook (Outlook only)
Press CapsLock and Shift back down to lower case (this is a per letter fix)
Fn key + Windows key may toggle problem on and off.
Alt key + Windows key may toggle problem on and off (especially in Win8).
Region and Language —> Keyboards and Languages —> change keyboards;
delete all keyboards other than your language and delete all instances of “Ink Correction”.
Disable Sticky Keys (Control Panel —> Ease of Access Center —> Make the keyboard easier to use —> Make it easier to type —> uncheck “Turn on Sticky Keys“)
Remove all keyboards from the Device Manager and reboot.
Login to the effected machine via RDP and then return to the machine and login locally.
Laptops may have to use the Widows key on the laptop and not a secondary keyboard.
In Win7 it may be useful to check for keyboard/language changing hotkeys here Control Panel —> Region and Language —> Keyboards and Languages —> Change Keyboards —> Advanced Key Settings
If you have only a left Windows key, you may attach an external keyboard which does. Failing that, user Matan below wrote a right-Windows-key simulator which can be found here. (Please note that I cannot guarantee the integrity of that source, so please use appropriate caution when downloading from unknown sources.)
User Klode as offered another possible solution for users lacking a right-Windows key. Open a command prompt and type first cd /d c:\tmp (then press Enter) followed by nircmd sendkeypress rwin (and press Enter). Klode also describes how to create a shortcut for that and his comment can be found here.