I think I know what kind of person I am.
The sentence above has an interrogative clause functioning as the complement of the verb know. The interrogative clause is:
This clause has a gap in it. The gap is understood as being co-referential with the interrogative phrase. The interrogative phrase here is the string what kind of person. So we could model the interrogative clause like this:
- What kind of personi [I am _________i]
- what kind of person [I am
what kind of person]
Notice that what kind of person functions semantically as the complement of the verb am.
This is very different from the following sentence:
- He chided me for being the kind of person that I am.
Here the relative clause that I am is modifying the noun phrase the kind of person. The reason for our using the word that here, is that it is introducing a relative clause.
Because the string I am in the Original Poster's sentence is part of an interrogative clause and is not a relative clause in its own right, there is no possibility of using the word that before the string I am. The following sentence is ungrammatical:
- *I think I know what kind of person that I am. (ungrammatical)