The first point is that verb ‘знати’ does have forms with prefixes: ‘ви-’, ‘у/в-’, ‘за-’ etc. So does almost any verb. Prefixes may change the lexical meaning of verbs and you should refer to a dictionary for a verb with prefix that you don't know yet.
It's true that with help of prefixes you generally turn imperfective verbs (дієслова недоконаного виду) into perfectives (доконаного). In your example: ‘знати → взнати’. And not only prefixes take part in this. You may also switch aspect (вид) of verbs by applying suffixes, shifting stress, altering sounds and replacing one root with another: ‘зна́ти (imp.) → взна́ти (p.) → взнава́ти (imp.)’
Important question is whether both verbs make a pair (видова пара) in which they describe the same action, except that one expresses imperfective aspect and the other expresses perfective aspect.
So back to your question there are so called ‘single aspect’ verbs (одновидові дієслова.) They do not have their match of other aspect.
‘Знати’ is an imperfective verb that is not intended to get a result and adding any prefix to it changes its meaning (взнати – to learn to know, to find out) so it belongs to that group too.
There are quite a lot of such verbs. Some of them are:
Only imperfective: володіти, ворогувати, зимувати, зорити, марити,
мислити, намагатися, потребувати, поважати, прагнути, сподіватися.
Only perfective: заплакати, наговоритися, надивитися, натерпітися, пропрацювати, стрепенутися, схаменутися, розгніватися, розговоритися.
I had known that...
cannot be expressed; it will sound the same asI knew that...
. It would help if you could put context around the words - examples of statements you'd like to express.бути/був
withбувати/бував
. The latter sounds much more "imperfect" than the former. But then, yet another affix, and…побував
, which is clearly perfect "have been".