How to express "Now I understand!" in Chinese

Once, one of my language partners asked me a question about Chinese. After I had explained, he replied in Chinese: "我知道"。I got confused, as to whether my explanation had answered his question, or I had failed to offer an appropriate answer he needed [1].
So, the next time, when [2] you want to express the meaning of “after you have explained it, I understood”, you should say “我知道了”.
In Chinese, the reply “我知道” means before you explained, I had already known it, so the way which is sort of impolite. We just would only say it to a person we have known know very well.
By the way, some foreigners tend to reply “我明白”. It’s ok okay [3], also but again, “我明白了” is better. However, “我知道了” is more natural.

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[1] In English grammar, there is a distinction between a definite article ('the') and an indefinite article ('a' or 'an').  We use 'the' when we refer to a specific something:

-- "I like the dress Sue is wearing."  (You are referring to a specific dress: the one Sue is wearing.)

-- "I live in the house with the red door."  (You are referring to a specific house: the one with a red door.)

We use 'a' or 'an' when we refer to a general something:

-- "I would like to buy a dress."  (You are referring to a dress in general.)

-- "I live in a house."  (You are referring to an unspecified house.)

So, let's look at a specific answer, and a general answer:

-- "I had failed to offer the answer he needed."  (You are referring a specific answer: the answer he needed.)

-- "I had failed to offer an answer."  (You are referring to a general answer.)

[2] Your grammar here was fine.  The following is fine:

-- "So the next time, when the lights go out, use a candle."

However, the above is a little disjointed.  The following is more natural:

-- "So the next time the lights go out, use a candle."

[3] The full word is 'okay'.  The short version must be capitalized as 'OK'.

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By the way, non-native speakers are taught that 了 means an action has been completed.  (The particle 了 has many uses, but this is the first we are taught.)  For example:

-- "我看这本书" means, "I am reading this book."  (The action has not been completed.)

-- "我看了这本书" means, "I have read this book."  (The action has been completed.)

For this reason, my understanding of your post is that:

-- "我知道" means, "I (already) know."

-- "我知道了" means, "I hadn't known this, but now I know this." Or, "My understanding before was incomplete, but now it is complete."  Or more simply, "Now I understand."