Context: Tenma Gabriel White 天真=ガヴリール=ホワイト has realized what she really is.
{sukui you no nai} dame tenshi datte koto ni ne!! 救いようのない駄目天使だってことにね!! A {[hopeless]} failure of an angel is [what I am]!
sukuu 救う To save.
sukui you no nai 救いようのない No apparent way of saving. Without a way to save [someone]. Futile to try to save. Hopeless. Gone for good.
ryaku shite datenshi'! 略して駄天使っ! Abbreviate it: failen angel! (this is Crunchyroll's translation, which I must admit: was great.)
datenshi 堕天使 Fallen angel.
daten suru 堕天する To fall from heaven. (i.e. to lose one's holiness.)
ochiru 堕る To fall. To drop. (in the sense of: to degenerate.)
The pun replaces the da of datenshi 堕天使 with da of dame 駄目, implying the angel isn't fallen become it's become evil, it's fail-en as it's become an useless failure who plays MMORPGs all day and has no future.