I think that eye contact is, if not an invitation to a conversation, at least a good signal that, should you approach them to strike one, they won't be startled. They know you exist now. Their first awareness of your existence won't be when you're breathing down their neck.
What I'd probably do, is try to find a seat somewhere close to her--maybe a table away, so she still has plenty of body space, but you're close enough for sociable interaction. (I know this situation is done and over, but this is for next time.) Then, from your strategic position, interrupt her absorption with her book using your voice. I wouldn't know what to say, but guys have used this approach on me before, at the library, and it's always worked to get my attention in a clear and genial way. One guy, strange enough, had this little--I think teapot--of coffee, and two tiny teacups, on the table where he was sitting. He got my attention by clearly saying, "Would you like some coffee?" I refused at first, laughing, and not believing that some stranger was offering me coffee out of the blue, but he must have been a good salesman, because he got me to drink his coffee, and I liked the chat we had.
It's almost like he had two teacups for exactly the purpose of inviting people to try his coffee. (And yes, I know, women shouldn't accept drinks from strangers, but there were an array of factors that ensured I was totally safe.) It achieved one thing that was probably helpful in starting a conversation, namely, that it got me to get up and walk over to his table.
I think, most of the time, everyone really likes being approached. If you're an amiable guy and just want to have a little conversation, I think that can't help but come across. I really doubt it would make her that uncomfortable, you approaching her. She might be surprised, but surprise quickly gives way to interest. In my case, anyway.