Grumbler, your translation makes it sound like he tried, but couldn’t log in. Failed to means that he never tried, he just didn’t do it. 

I thought about this, but in combination with "to play game" it makes no sense to me. Does this mean that he could play either with or without logging in?

It sounds like he was supposed to play the game, but he failed to login and do it. It’s a strange example. Maybe they had arranged to play beforehand.

Переделал - с большим трудом

I think it’s accurate in view of the original. Perhaps the author will honor us with an explanation.

cool