I need help choosing a fixed gear bike [fixie] for a person who is 5'1"?

So for my girlfriends bday/anniversary/x-mas gift. I want to buy her a bike so that we can go riding together. She really likes fixies since she had one before. I have no clue as to what to get her. She is 5 feet and 1in tall. It would be awesome of the bike cost less than 400$

Thanks =D

Or you guys can just give me some recommendations on fixed bikes that you think rock and/or forums on marketplaces w/ fixed bikes and I’ll go look