After a suicide attempt, a girl meets a man in the psych-ward, claiming to be a captured superhero; she becomes his only hope for survival.

This is much better than your first logline post but, like Nir says, the final clause in your sentence needs to evoke a sense of what's at stake

Hospitalised after a suicide attempt, a girl meets a man in the psych-ward who convinces her that he has super-powers and needs her help to escape certain death at the hands of his nemesis who wants to absorb his powers and destroy the world.

The second draft is better but lacks the setting.

Who is the main character the girl or the superhero?

Why must she save the super hero? What is at stake if she fails? Why can't the super hero save him self if he is indeed a super hero? Or is he an inmate who believes he is a super hero and not a genuine one?

I think if she falls in love with him then she will be better motivated to help him and the stakes are higher. In order to be with him on the outside they both must get better and she makes it her mission to get him and herself better.

Hope this helps.

I think this explains the plot better:

"When a suicidal girl meets an inmate calming to be a superhero, she must save him before his nemesis can gain control of his powers."