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What do you think a good next approach for her should be? Just something easy like "Hey, sorry I was rude when we first met, I forgot to ask you if you wanted to grab a drink after work sometime?" Like this or is it too straightforward?
Sounds decent. Maybe say, "I thought you really enjoyed talking with me before. Which is why I was suprised you didn't ask me if I wanted to grab a drink after work sometime." Basically you switch the roles around. You could also try, "If you want to go get a drink with me, we could try to figure out why we look fimilar to eachother. But the offer is on the table for today only, I am a busy man you know."
Also I think it was better you didnt ask her for her number when you first met her. You got her thinking of you and now she will be on the look out for you, which is better in the long run.