This is a great question, and it led me down a bit of a rabbit hole. This kind of clause is called a Gapless Relative Clause. The sentence could be written as you have it, or with "I don't know what it is" - the "it" is called the Resumptive Pronoun which are "common in spoken English but are officially ungrammatical".

The Wikipedia article has a similar example:

In other cases, the resumptive pronoun is used to work around a syntactic constraint:

They have a billion dollars of inventory that they don't know where it is.

In this example, the word it occurs as part of a wh-island. Attempting to extract it gives an unacceptable result:

*They have a billion dollars of inventory that they don't know where ___ is.

Here's another great article I found which sums it up well:

"Resumptives are non-standard, but in such cases they're much better than their gapped counterparts, which people usually find incomprehensible, or at least very hard to comprehend."

So basically, your original sentence is "unacceptable"/"incomprehensible", but adding "it" would be grammatically incorrect but easier to understand. Best bet is probably to totally rephrase the sentence as others have suggested.