Question: How can I offer help in a way which
a) Would make it likely for her to accept if the need exists.
I think serious topics like this are often served well by having a direct approach, and since you mentioned that you don't know her extremely well, it may go over best if you include in your offer an easy way for her to reject it. Something like: "you mentioned the other day dealing with some depression. Pardon me if this is uncomfortable for you or if you're dealing with it some other way, but I have been through a similar situation and I'd be willing to talk about it or help if you feel that would be helpful. For me, having someone to talk to has helped me in the past." (Adjust so it's accurate for your situation.)
I think a proposal like that is direct in that it makes it clear what you're offering and why, and since this might be inherently a touchy or uncomfortable topic between the two of you offering easy way outs can communicate that you aren't trying to be overly invasive or pressure her into confiding in you.
b) Would not be perceived as a romantic approach (I am a heterosexual male, she is single and attractive, the conversation started in a rather flirty way).
This is very nuanced and often has as much to do with the overtones of what you say more than what you're saying. Additionally, we often let on more of our intentions than we realize, so be aware that if you have romantic ambitions alongside genuinely wanting to help, there is some risk she'll perceive that and interpret it in her own way.
Of course, if you start talking about that, honesty is the best approach. It is possible to both be attracted to someone and be genuinely willing to help whether or not that attraction is shared.
At any rate, if that isn't an issue, just be straightforward and try to avoid innuendos. Also, be understanding of the fact that people come with inherent biases that color how they perceive interactions. For example, a non-trivial amount of people believe that men only become friends with women they find attractive, and vice versa, so try to be understanding if there are misunderstandings.
Depression is a tricky issue that not enough people are willing to discuss, so I think it's a great idea to try and help each other out. I certainly agree with you that having someone who genuinely wants to talk to you is helpful.