As for your bike dilemma, how much do you want to spend and do you plan on any other uses?

With a short commute as yours, any bike will do the job. the cheapest option would be to keep your mountain bike and do the following:
– Install relatively narrow high pressure slicks (something like 26" x 1.4 or 1.6", high pressure, and little or no thread) for about 9 months per year.
– For Winter, either install back your knobbies or get Nokian Hakkapellitta tires (studded tires; expensive).
– Install a rack and lights.

As for getting a new bike, you could aim towards a hybrid or a touring bike, which the provision they accept 700x37 tires for winter. But that would typically be if you want to ride your bike more often and under what conditions.