Forgive my shock and disgust that janitorial positions are apparently in need of formal resumes and cover letters. That said, I've only ever written cover letters for exactly the situation you don't want: addressing a specific hiring manager by name.
I'm not entirely sure how you're going to write a "generic" cover letter. I mean...what would you put on it? The whole point of a cover letter is either:
a) To simplify processing. The person on the fax machine might not be involved with hiring or resumes at all. And in large companies, there may be hundreds of job listings, so it is useful to identify which listing a resume in intended to be an application for. So, a quick "this is for person X in regards to position Y" makes it easier for the receiving company to figure out what to do with it.
But if you don't know who to send it to, or even whether the company you're sending it to has a distinct human resources department, this may be impractical.
b) In the case of companies who have posted specific job listings with specific requirements, to quickly identify that a resume submitter
has read those requirements, understands them, and does meet them.
But if you're not responding to a specific posting, you probably don't have a list of specific requirements to confirm that you meet.
I need help.
I think my advice in this situation would be to keep in mind the
spirit of intent for "a" listed above, and use a minimalistic approach. Simply include whatever information would be required to simplify the process of getting your resume from the fax machine, into the hands of whomever you want to see it. So...name and contact information, and a brief 2-3 sentences explaining the kind of work you're looking for and a polite request to please contact you. I would advise against duplicating information from your resume. Make the cover letter a processing and delivery tool, not an advertisement.
Write only what you need to quickly and efficiently communicate your intent, and to make it easy for whomever receives the resume to give it to the right person.