Is there anything wrong with this (details inside)

You can consider this part 2 of this question, as they both have similar background stories.
OK, so there was another writing assignment. I don’t remember exactly what it was but I remember at one point I wrote: ”...I tried to persuade him, but he said that that was impossible”. When my friend looked into my paper, she said that I couldn’t use the word “that” twice in a row like that, it was grammatically incorrect. I tried to convince her that the two words “that”, despite being so close, actually had two separated meaning so it was OK to write like that, but she denied what I said by stating: “You can’t do that. That’s what they told me at the language center”. Eventually I gave up, mostly because I got skeptical about my knowledge after hearing her statement.
I still can’t figure out if I was right or not. Can anybody here help me?