Jane and I are friends. (I know / I've known) her very well.
The answer is 'I know'. I want to know why 'I've known' cannot be the answer in this short question.
"Jane and I are friends" implies present tense.
"I know her very well" is also present tense, while "I've known her very well" implies present perfect tense. The correct grammar matches both tenses of the two sentences. Therefore, "I've known her very well" is incorrect.
If you wish to use "I've known her very well", then I believe "Jane and I have been friends. I've known her very well" is grammatically correct.
To further clarify, the conjugation "I've" can be broken down into "I have", which implies present perfect tense, or ownership.
Present perfect Tense: "I have [verb]" This expresses the idea that something occurred at an unspecified time before now or possibly even now. It's similar to saying "I have the experience of..." For example: "I have walked with her before"
Ownership: "I have [object]" For example: I have an apple.
I hope this helps! Best of luck with learning!
EDIT: Replaced "past tense" with "present perfect tense". Explained a bit more what present perfect tense is.