"I know that."
It could be my bad Japanese in feeling that わかります is acceptable, but わかっています is definitely better since we are understanding and currently still understanding.
From what I understand, psychological verbs such as 知る and わかる generally take the ~ている form when you are in that state (I am in the state of knowing, for example). Other verbs like 住む just take the ~ている conjugation in general use since it makes more sense to talk about the resulatant action than the action itself (住む means to live at a place, so it would be common to talk about where you are currently living at).
The English verb "to know" can mean different things, depending on the context, and it's not uncommon for other languages to use different verbs for these different meanings.
If you think of Maya Angelou's "I Know Why The Caged Bird Sings"; plenty of people know that the caged bird sings, which just means being aware of it - but not everyone knows why, which implies a deeper level of understanding.
So yes, わかる does mean "to understand"; but in some cases, that is what's really meant when someone says "I know".