After uninstalling and installing the Nike+ Connect app in multiple ways, I finally found the cause of the problem and a way to fix it.
The problem is in the Kernel Extension "/System/Library/Extensions/NikeSportWatch.kext". In Console.app I found several messages that said "NikeSportWatch.kext has no Info.plist file.".
With the Terminal.app I went into that directory and indeed there was no Info.plist file, only a info.plist file. So I renamed that info.plist file to Info.plist file and executed the command:
sudo kextload /System/Library/Extensions/NikeSportWatch.kext
And after that, as soon as I connected my watch, the Nike+ Connect software finally recognised my watch!!
The volume where OSX is installed in my case is case sensitive, but by default it's not. This might be the case for other people in this thread too?