I'm guessing you already checked the settings to see if all looks ok there:
Settings > Language & input >
My Father has the Droid Turbo and uses the speech to text a good bit when ending SMS etc. and isn't having an issue.
Yes. Enhance voice is selected. Tried simple voice just to see but it still doesn't work. The little microphone is there, it just doesn't do anything when you click on it.
Strange. Maybe try to plug it into a computer and run the Software Repair Assistant. Select the option to backup your data. Its worth a shot. Here are the steps. I've used the tool helping family members and it has worked on various issues we have had.