How do you enforce handwriting standards for homework assignments as a TA?
My approach is simple- homework is submitted as a typed .pdf file, no exceptions. Homework is submitted online through the course management system (Instructure's Canvas in our case), which enforces the required file extension.
Yes, this means that my students need to learn how to produce typed mathematical equations. They're welcome to use LaTeX (probably the best choice in the long term), Google Docs (which has a pretty intuitive system for math formulas that's adequate for most purposes), MS Word (which has an awfully complicated and hard to use equation editor), Apple Pages, or whatever software they like, but they have to convert the output to pdf so that I don't have to deal with all of these different word processing software packages.
Make a pile of homework submissions that are not legible enough for you to do a good job of grading. Put those in a separate envelope in the professor's mailbox, along with a cover note explaining that you were not able to grade those.
Let the administrator in charge of TAs in your department know about the problem.
If you are given responsibility, and authority, to address this problem as you see fit, then I would suggest paperclipping a quarter sheet to each illegible submission, with a xeroxed note, letting the student know that the temporary grade is zero, but that if the assignment is resubmitted in legible form by (deadline), you will grade it as though it had been submitted on time. Explain that if a student is unable to submit a legible version in writing, s/he can see you in office hours to use an alternate method of demonstrating that s/he did the work. Keep a record of which students submitted illegible homework. Use this list to work toward getting the homework submitted legibly and on time.
The homework is worth 5% and the professor wants to give full grades to anyone who completes it. I'm basing my answer on the assumption that the students can get their graded assignments back after you mark them.
If that is the case, explain to them that for you to give adequate feedback, you have to be able to read the assignments.
The assignments aren't worth much, so I imagine the professor is giving them out as an opportunity for the students to make sure they understand how to do the problems assigned. Although it may not directly change their grade, your ability to understand their answers will change the quality of feedback you can give. Explain to the students that if they want constructive feedback, they must write neat enough for you to understand.
The professor doesn't seem to care about giving that 5% away, so you shouldn't worry about it either. Legibility of the answers isn't as much for your benefit as it is for theirs when they receive comments back on their work. If they don't make it legible for you; then it's just less thought that you have to put towards giving feedback (since you can't give feedback on answers you can't understand). That way you aren't wasting your time giving feedback to students who don't really want it; and those who do want it will do a better job.