I corrected the student, saying that he should write "How much water do you use to take a bath?" because his sentence seemed unnatural to me. Do you consider it correct? Would you use it?
The question in your post is correct: “How much water do you use to take a bath?” You could also say “How much water do you use for a bath?”
The question in your title (“How much water do you take a bath with?”) does not sound good to me: you take a bath with bubbles, with a magazine, or with a dear friend, but not really “with water”.
I think your student's formulation is grammatically correct. Some people find ending on a preposition to be stylistically infelicitous; the use of "with" refers to an item (the water) that came earlier in the sentence, which also makes it a little awkward. Perhaps that tweaked your sensibilities.
There is also a connotative issue - in this context one bathes with someone or something in addition to the bath. Obviously, because the object is water, one understands that the water is constitutive of the bath, but still.
"How much water do you use to take a bath?" is perhaps preferable because the object comes first (and that is what the question), AND it lacks the anaphoric structure of your student's sentence, AND it lacks the slightly odd connotations of "with" in this context.
Edit: I should add that I probably wouldn't have picked the student up on this. The student's usage isn't especially jarring to my ears, and if anything sounds more natural to me in speech than your formulation. A more natural phrasing in speech would be rather shorter, such as "How much water do you use for a bath"? I would also have no problem with "How much water do you bathe with" in speech, and in writing the only issue is the connotation of "with" in this context.
I would suggest that "how much water do you take a bath in?" would be grammatically correct if you don't mind ending sentences with a preposition. I don't. Thinking about it in terms of the answer to the question can be helpful here. "I bathe in six inches of water" would be OK, "I bathe with six inches of water" wouldn't make sense.
Grammatically the sentence is correct although a bit awkward. It's a case of a sentence ending with preposition. Something like, On what did you step? or what did you step on? both are correct, although one with preposition on end is more natural. In your case it's other way round. "With how much water do you take a bath?" seems more natural than "how much water do you take a bath with?" I think it's good that you corrected your student but make sure to explain him the real reason.
The sentence in your title is incorrect, because it ends with a preposition. You can fix the sentence by moving "with" in front of what it's referring to, but although it would be grammatically correct to say "With how much water do you take a bath?", I think that sounds more awkward than your alternative.