____ did you manage to do it?
However is the correct one.
HOW EVER IS THE CORRECT ONE.............'CAUSE HOWEVER IS USED IN THE MIDDLE OF THE SENTENCE NOT IN THE STARTING OF IT.......!
If it was however, it would need a comma after it. To ask IF he did it not HOW he did it.
I picked How ever. It's like how did you ever manage to do it...?
How ever is the correct answer. However, others may disagree. How did I ever know it was correct? Well, however has a different meaning to "how ever". "However" would be correct if there was a comma before "did". Why? Because it would mean something different from "how did you ever manage to do it?". "However, did..." means did you manage to do it despite something else (as an example).
Both are correct.
However is used to express surprise: However did you manage to get him to agree to that?
How ever is used in questions to emphasize the question word: How ever did he manage that?