Sorry to say, this girl sounds like a trainwreck. At the very least, she has really low self-esteem. It seems that no amount of flirtation and attention from men is enough for her. And I think it's a little weird (and rude) to be on a date with someone and be talking about trying online dating.
Does your company have a policy forbidding dating coworkers? If so, I'd be really careful here--she could turn on you and claim sexual harassment. I don't mean to sound paranoid, but it could happen.
I have to agree with FG52. be careful, walk away. she does seem to have personal emotional issues. that added to a potential trap situation of dating someone you supervise, there is just too much risk as compared to the potential gain.
You know the famous saying " Don't c*&p where you eat." well from the sounds of it, she could really cause damage to you emotionally, and career wise..
its a permissitic view but worth considering.