No one's answered your question in over a year?! I found the answer in Danish: An Essential Grammar.
It calls object pronouns (like ham (him), hende (her) and so on) "light elements". Light elements are unstressed and in a normal sentence and come straight after the verb.
Jeg kender ham ikke (I don't know him)
Jeg finder hende ikke (I don't find her)
If you want to stress these light elements, then they move to the normal position you'd expect them in, after "ikke".
Jeg kender ikke ham (I don't know HIM)
Jeg finder ikke hende (I don't find HER)
Now you've got the stress on "him" and "her".
You can do the same with words like her (here) and der (there).
Hun er der ikke (He's not there)
Hun er ikke der (He's not THERE)
This sentence doesn't make sense. "I am not finding her" and "I don't find her" aren't used as sentences on their own in English. You only say "I don't find her" in combination with an adjective, to express an opinion about someone, e.g. "I don't find her attractive".
If this sentence is supposed to mean "I can't find her", then I have to disagree with the currently accepted translation.
About the 'It would have to be "Jeg kan ikke finde hende" to mean "I can't find her"': I suppose in this case it doesn't make a difference whether you add the "kan" or not, it will always be translated to "I can't find her". It's the same in German: "Ich finde sie nicht." and "Ich kann sie nicht finden" both mean "I can't find her."
I agree that it's a very unlikely sentence, but I wouldn't say that it's impossible to use the sentences "I'm not finding her" or "I don't find her" on their own in English.
"She's gone again; go find her." "No. I'm not finding her. You do it." "I don't find her, you find her. It's your job." "Fine. But I find you really annoying. Just so you know."
It is not supposed to mean i cannot find her it is supposed to mean i am not finding her. It is almost but not entirely the same.
Jeg kan ikke finde= i can't or I am not able to find Jeg finder ikke = i am not finding (for whatever reason) maybe i will find her in the future.
Most often you would also say I can't find her, jeg kan ikke finde hende, in danish.
Didn't know that song, so had to look it up. Pretty cool (:
As to your question, in the song they sing: Jeg har et lille problem jeg ikke kan finde før du viser mig hvor du gemmer hende
This would be: I have a little problem (that) I can't find before you show me where you are hiding her
So he has a problem that he can't find until it (being the city of Copenhagen) shows him where it is hiding her.
Hope that helps!