A number of students requesting a number of reference letters. What to do?
Your director says "As many as you can" - in my philosophy these would be those where I think the letter would easily help and, as such, easily written.
Create a spreadsheet with comments, and evaluations, possibly link collections if that's easier than to search through mail.
Use automated template generation for the reference letter outline, possibly electronically added signature, and write short, crisp paragraphs on what impressed you about the student. One, at most two paragraphs are perfectly enough if you have something good to say. It's like feedback for an essay. "This student performed well because they ..., were able to ... and showed skilful handling of ... Whatever is relevant, give facts and a brief evaluation, this is perfectly fine.
Any awards, just mention them.
End on a standardized positive note. This is not a tenure reference. If you have to write many references, you cannot spend much time on refined formulations.
Group the references together for the same website/uni and fire them off.
For very strong students, usually it is possible (and pleasurable) to write more, as you will know more about what they have done. They can be treated in a more individualized fashion.
References for weak students, on the other hand, are very difficult to write, and so they will burn a lot more of your time if you want them to help at all; this will be like squeezing blood from a stone. I do not recommend doing that, unless you have some simple and strong thing to say about them and you wish to do so.
Although it's not up to you to decide whether they should proceed with a graduate education or not, if you yourself do not think they should continue, it is in your right to choose not to be part of the process in that case.
This is complementary to the answer of Captain Emacs.
For those students for whom your only contact is a class or two, tell them when they ask that your letter will need to be fairly pro-forma, describing the course and how they did in it. Tell them that other writers may have more valuable things to say about them. Then ask if they are still interested.
The best letters come from those with the most contact, I think. That may not be you. Make sure they understand that "full marks" may not be enough to judge future success.
This might cut the numbers down a bit and also permit you to give honest appraisals as far as you know about their likelihood of success.