In your example the inner clause is a full sentence that could stand alone as a complete sentence:
Ich kenne dich.
I know you.
You also can say, that you know yourself:
I know I.
Or, lets say, you know someone else, a male person:
I know he.
What do you think: Are these English sentences right or wrong? Due to your own argumentation they should be right. But they are not. In fact you say in English:
I know me.
I know him.
Also
I know she her.
I know we us.
I know they them.
English has almost completely lost the system of grammatical cases that the common ancestor of English and German (and some other languages) had, which is Proto-Germanic, or even before that: Proto-Indo-European.
In English grammatical cases survived only in personal pronouns, and there are only three cases left:
- nominative case
I walk. He walks. We walk.
- genitive case
This is my house. This is his house. This is our house.
- oblique case
The house belongs to me. The house belongs to him. The house belongs to us.
They watch me. They watch him. They watch us.
In case of 2nd person (you) nominative and oblique case by chance are identical:
- nominative case
You walk.
- oblique case
The house belongs to you. They watch you.
The old system of previous eight grammatical cases in Proto-Indo-European also did not fully survive in German, but it is much more alive in German than in English. In German we use it not just only for pronouns, but also for nouns, and we have 4 cases, where the English oblique case matches with two different German cases, which are dative and accusative case:
- nominative case
Ich gehe. Du gehst.
- genitive case
Das ist mein Haus. Das ist dein Haus.
- dative case
Das Haus gehört mir. Das Haus gehört dir.
- accusative case
Sie beobachten mich. Sie beobachten dich.
So, when you compare the English sentence with the German sentence, you will notice, that in "I know you" the word you is in oblique case. And in this very simple example you can do a simple word-by-word translation, because the grammatical structure (subject, predicate, object) is the same in both languages. (You normally can't do such a simple word-by-word translation, but here, in this example, it works.) But then in the German sentence the object that was in oblique case in English now has to be either dative or accusative case.
To decide which of both you have to take, you have to look at the verb. Look at this question for more details: How to know Nominative, Accusative, Dative, Genitive Verbs in German?
For the verb kennen the object must be in accusative case. So it has to be
Ich kenne dich.