Hi C&B, your sister is lucky you've allowed her to stay this long. I'm not sure why you think she'll be homeless if she works. Surely she can pay enough for a room somewhere?
I've seen others on these forums deal with the same situation by giving their relative notice that they'll need to move out by a certain date. You can be generous time-wise, but be careful never to back down, not one inch, so she knows you mean it.
I'd be careful of raising a 3yo with a drunk; it's not a good example.