Because he, obviously, is in the process of it, right? otherwise you would not have the 'again' in there. And as such you need to have continuous tense
It is not so literal - a specific verb vs. to be and an adverb. But since we have no direct Czech equivalent of asleep, I will add it. We do have vzhůru for "not asleep" in Czech and we have "spící" for "sleeping".
However, if your husband has had a long period of troubled, sleepless nights, but now he finally is sleeping normally again, I think one would indeed say "My husband sleeps again". (I think - non-native speaker though.)