Consider that she might be feeling as awkward about the situation as you are -- but you'll be at an impasse until one of you decides to do something about it.
Walk right up to her cubicle, tomorrow morning if you can, and say this:
Hi. I was feeling that it's kind of weird that we work next to each other and we've never been formally introduced. I'm Duncan (extend hand to shake)
Simple, quick, done. No more anxiety.
After that, just let the conversation take its natural course. If she brings up the LinkedIn request, you can be honest and laugh about it with her:
Yes, I remember. Actually that's why I was feeling even more awkward, since we're connected online and we've never actually connected in real life. It's a strange world, huh?
Ideally you'd wait for a moment when she is not interacting with other people, but if no such opportunity presents itself, then I'd say something like,
Sorry, I don't mean to get in the wayinterrupt anything. I just wanted to take anthis opportunity to introduce myself as I've been here five months and it's kind of strange I've never actually officially spoken to any of you.
Then introduce yourself to everyone around her, or at least those who are paying attention and who don't seem too busy.
I just noticed your comment that "she is attractive and around my age" which suggests you have some romantic interest in her, and which is probably the source of some of your discomfort. Dating coworkers is, well, complicated, and while it's not unheard of (my brother ended up marrying a woman he met at his first job out of college) these days it can quickly get you into trouble if you misread the signals.
Given that, I recommend at the very least pretending a lack of any romantic interest when you first introduce yourself. See how she responds to that, first. You should probably also check if there are other answers here to the question of how to ask a coworker out on a date.