@PhilippeD. are you using FAT32 filesystem? Your computer would see NTFS, exfat on the Kobo fine and able to copy files to, but the Kobo itself would not be able to access those files because it doesn't know the filesystem. It needs a single partition, with oldschool FAT filesystem on it.
I think I've read something about a bug with microsd handling recently... which firmware version are you using exactly?
@Luffy: Does it work on any other PCs? Anything with Windows 7, Linux, ...? The Kobo itself does show it's connected to PC? Do you have antivirus, such things sometimes interfere too, temporarily disabling might be worth a try. You could also try simply formatting it but that would lose you all your data. :-/
Is your Windows updated? I thought the detection bugs we had a year or so ago were fixed
