Sounds like you have great groundwork for a genuine conversation. Send a message that says something like, “Thanks for accepting my connection! The work you’re doing at Microsoft seems fascinating, and I’m especially interested in [project they are working on].” Then ask a question that could lead to a conversation. You could ask about their favorite part about working with a specific tool, or if they think X or Y is better for Z. You could tie it into the types of comments they have been responding to in your feed.
The trick I have heard is, after an exchange or two, ask what the biggest challenge their team is facing. Then using that knowledge, design some kind of pitch of how you could address that challenge with your skillset. You can go back to her, and say you were thinking a lot about that challenge, and you had some ideas. Supposedly, she should know they have an open spot, and refer without you even asking! But failing that, you could mention it if she doesn’t.
One thing to note about Microsoft, they have a specific referral process where you are entered into the system by a recruiter, you can’t apply as an external applicant and be referred after the fact, unless things have changed since I launched their recruiting software ten years ago. Your referral will likely end up being a new application, but I think that will be fine.