From the description, it sounds like you are using Windows. Which version? And is it up to date? The description fits the problems with Windows 10 discussed in
Calibre not seeing Kobo in Windows 10 Preview builds. But, if are using Win10 and have everything up to date, then it should be OK.
Any chance to try a non-Windows computer? A Mac or something running Linux should be good to prove whether it is the device.
An alternative way to get books on the device (other than buying from Kobo), is via the browser. If you are using calibre for ebook management, you can enable the server and use the browser to look at the book list and download the books. It isn't perfect, but it lets you get something to read while sorting out the real problem.