I'm going to make someone walk in cold weather so I won't be late for work.
Why don't you call your boss and let him know you have an emergence to take care off?
Its not like I enjoy making her do that but I can't make it to work on time if I drive her first.
Next time you need to leave earlier so you can drive her to her destination while still having ample time to make it to work. It is unfair to her if you don't have good time management skills. Try to work something out because it's just a small-time constraint issue that can be easily rectified.
I'm not apologetic about it, nor rearranging my life to do her a favor. She's no friend or relation of mine, and it's not my fault she doesn't have anyone else to ask.
It's true, you're not obligated. Knowing her situation as you do, it would be kind, though.  If you were in her situation, would you want someone to help you out, even if they weren't obligated to do so?