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.. from everything in the letter that the obvious answer was that the LW had no chance, but then comes that last line about her readily accepting drinks, so assuming that they don't work in a bar, she is apparently going on mini-dates with the LW after work (or is this in a group setting?). So what is that all about?
I think if I was the LW, I would tell the lady that I wanted her to have my babies too, and see what she said. Probably she would laugh it off and say that it was a no-go. But either way the LW would probably get some valuable feedback about her intentions.
If I want to flirt with married ladies, I will often start by asking something like So how often does your husband beat you, then? to get a feel for the state of her marriage.