Actually 'plejer' also means 'caretaker' (I think that would be the closest translation). If you say: "Jeg plejer ham", you could translate it to "I take care of him"
So 'sygeplejer' literally means someone who takes care of the sick, but I understand why you find it funny :)
Actually that 'sygeplejer' is male is kinda right, but mostly wrong.
In the olden days 'sygeplejer' refereed to a male, and 'sygeplejerske' to a female performing the same job. Danish used to do this with a lot of words, just like German still do, but for this one both words stuck.
In modern Danish though they are actually two different jobs. 'sygeplejerske', the female form, is actually a job requiring a higher education then a 'sygeplejer', the male form.
In even more modern times, since 1991, the education for 'sygeplejer' no longer exist and has been replaced by 'social- og sundhedsuddannelserne'.
Therefore 'sygeplejerske' is definitely a correct translation and, I would argue, the only correct translation. Also if you look up 'nurse' in a English to Danish dictionary you find only 'sygeplejerske'. https://www.ordbogen.com/opslag.php?dict=a000&word=nurse