"She plays piano while she sings songs."
Translation:Hun spiller piano mens hun synger sanger.
As in "She plays the piano while singing songs"? No, I can't find a way to say that and keep "mens" in the sentence. You need the object as mens ties two complete sentences together.
'men' and 'mens' have different functions. 'men' can be followed by a verb in a few cases, but not usually.
"(...), but isn't that weird." = "(...), men er ikke det rart."