You do not need drivers for speakers and it doesn't matter if your computer detects the speakers as headphones, it outputs the same analogue signal regardless. If you can plug headphones into the same port and they work but your speakers don't, you have a set of broken speakers or a broken audio cable. Reinstalling Windows shouldn't be necessary, you will be able to manually download speaker drivers if your computer has not done so. Type "Device Manager" in the Start menu search bar and look under "Sound, video and game controllers". Your sound controller will probably be called "High Definition Audio Device", double click it, go to the Driver tab and click "Update Driver...". Windows should be able to download the driver automatically.
Also ensure you don't have anything plugged into the front port, the sound card likely has a hardware switch that prevents audio being outputted when the front port is in use.