[SomeCallMeChris]

I am only an intermediate student, so any corrections to my Japanese by fluent students or native speakers would be welcome.

For an all-kanji phrase that has a close meaning, I would use,
不寝番天使

'Sleepless angel(s)' or 'Unsleeping vigilant angel(s)'.

Or even the originally suggested '不寝天使’　

Either way, making a compound like that suggests that you're creating a noun to describe a group of angels that don't sleep, and it's not a complete sentence, ... and it's somewhat coining a new word. It also might be more suggestive that there are angels that are not sleeping now, rather than angels that do not ever sleep.

To say it properly, I would use,

天使は寝ることがいらない。

That's literally a little more like 'sleep is unnecessary for angel(s).'

If you want to make it quite clear that you mean angels as a whole and not a particular angel, maybe,

天使の皆は寝ることがいらない。


If you want to be much more literal, 'Angels do not ever sleep' might be,

天使はぜんぜん寝ることがない。

Maybe you can use 眠る in the same way as 寝る in these sentences, but I'm not so familiar with using 眠る.