I'm 5'8" and I currently ride a 54cm frame. Although, to be fair, it is simultaneously too big and too small for me, in different areas. But that has to do with my leg, arm, and torso lengths which are "non-standard". I've made it work, but I think I'll eventually spring for a custom frame.

Anyway - probably a tad small, but I bet you can make it work while you find something that fits better. I'd recommend going to see a bike fitter. I put it off for 4 years, and finally went to get a bike fit. Very educational. A professional bike fit is a very good thing.

And your feet don't need to touch when you are in the saddle.