Resume being passed along, including cover letter

Yes, definitely include a CL. It highlights your experiences off your resume.

No, don't send in a cover letter, they're anachronistic to begin with, and sending one once you've already done what it's supposed to do it stupid.

You could always ask her...

No unless he/she is requesting a CL.

A good cover letter will only help you in a marginal way but a bad cover letter can completely screw you and many college students write mediocre to bad cover letters IMO.

However, you should drop a few quick lines in your email when sending your resume, basically to thank him/her for forwarding your resume to the appropriate contact for the consideration of the SA/FT position.

Cover letters generally don't help nearly as much as people assume they do. That said, if they are needed, not having one will definitely hurt you. If I were you, I would send the resume over as soon as possible, and then in the email besides thanking her, just ask if her bank wants a cover letter as well. If not, she'll let you know not to worry about it - if so, she'll probably say something like good point, send that along as well. Don't overthink this - get your resume into her hands first and foremost, thank her, and then just ask if she needs a cover letter as well.

As a separate note - although it should go without saying - absolutely check and recheck your resume for any silly errors in punctuation, grammar, structure, format - it doesn't take long, but is necessary.

Good luck.