How to get a job at Dr. Offic when they are not hiring

Hi, If anyone can give advice I would really appreciate it. I need a job. I just got my LPN and can't work in a hospital or nursing home because of the hours. I'm a single parent and using all off help to attend night classes. I would like to work in a Dr.s office but no one is hiring in my area! I have heard that these positions are usually found through other people. I need a part-time position that is sort of flexible because I go to school at night for ADN. Is this even possible??. I cannot work at my current position anymore because I refuse to continue to do what I'm doing after I've worked so hard to get my LPN. I thought I would fax my resume to a couple of places . What would I say on a cover letter to a place thats not even hiring? Or maybe they are hiring and they are not advertising? Does anyone have a suggestion on how to go about doing this in a professional manner without looking desperate for a job??