Speaking as both a professor and corporate division manager at different points in my life: Your cover letter is not your CV, and (as said by ItWasLikeThat) you should not try to cram your CV into it. The cover letter is more of a polite marketing tool to convince the professor you are the one for the job.
So I am interested in a brief summary of your experience and education, your GPA, courses you took specifically related to MY project / business.
More importantly, why are you interested in MY project / business? (and incidentally, this will show me if you have done your homework to read anything about me and my project / business).
Related, what do you hope to learn working on MY project?
I have to reject applicants; and although your CV will play a factor, your cover letter should be telling me why you should be on the short list. I would rather have an intern interested in my project and wanting to help it succeed than an intern that spends every minute alone staring into their phone and typing to their friends.
Obviously it must be short, you must mix relevant experience with your motivation to contribute tangible results and learn what the internship is supposed to provide to both of us; practical (i.e. real life) work experience for you, in exchange for relief of some novice-level workload for me.
If possible I will choose an intern with actual interest in my field; and that will override some CV hiccups (like your 'C' in art history) and whether you are in a barbershop quartet, on the soccer team, President of the Student Council or volunteer at the soup kitchen.