What's the difference between 知ります and 知っています here ? Why can't we use 知ります or 分かります？
I just wanted to add that while ～ている can have the meaning of being in the middle of an action (like eating), it is also used to explain a circumstance or condition. Another verb that acts likes 知る(to know) is 結婚する(to marry).
For example, if you say 私は結婚してる, it translates to "I am married" rather than 'I am marrying'.
Cautionary note: You will almost never use 知る in its positive dictionary form. When you want to say that you know something, always say 知っている. However! The tricky thing is that when you don't know something, you should say 知らない. Nobody has been able to explain to me why this is, but you should still keep this nuance in mind.
But you can say 結婚していない = I'm not married. Go figure :P
＞ 知っている and 知らない
You raised a good point. I didn't think of it before. I guess it has to do with whether the circumstance/condition can be regarded as temporary or not.
If you use ている (or てる), you're aware that the condition was different before, but now it's in the current condition. He didn't know it before, but now he knows it = 彼は知っている. The speaker wasn't married before, but now he/she is married = 私は結婚している.
If you think of a specific knowledge as common sense, or can completely ignore the situation where one doesn't know that fact, it is quite common to use 知る. For example, "A scientist whom everybody knows." = 誰もが知る科学者 (It is perfectly natural to say 誰もが知っている科学者, though).
On the contrary, when one doesn't know something, he/she wouldn't even be aware of a situation where he knows that knowledge. There's only one permanent situation - not knowing - 知らない. (Wow it's getting philosophical). Here too, if you're discussing of a knowledge that one doesn't know but is expected to know in the future, you could use 知っていない - but even in this case it's usually 知らない, and I have to admit I can't explain why.
As for marriage, there could be, and you know that, there's a situation where you're married. Thus it's 私は結婚していない。がんばってね！
A bit confused on the last point. Do you mean only if I was married before but not married now, 私は結婚していない should be used?
No, you can use it even when you've never married (this applies to almost every form of ていない). You can still imagine then, that you might be married in the future, however unlikely it may seem to you.
From what I understand (I am still an beginner) 〜ています is used to indicate a present ongoing action, such as "I eat" = 「私は食べます」where as "I am eating" = 「私は食べています」. The first one just stating the fact that you eat sometimes. The second one states that you are currently eating.
So, in this sentence you currently know her family (are in a current well-informed, or so, situation with her family).
That's my understanding from a beginner's knowledge. I would appreciate correction if I am wrong. ;-)
P.S. Of course「私は食べます」could also mean "I will eat"