Is cafe an English word or a misspelling of café? The same goes for touche and touché.
This isn't the same as this since I'm asking if cafe is a English word, not if I should use a diacritical mark on the "e".
You are asking whether loanwords from French are considered English words in their own right.
According to Loanwords: major periods of borrowing in the history of English by Prof. S. Kemmer of Rice University, loanwords generally go through the following process:
It is implied that loanwords are English words in their own right, based on indisputable examples "of the loanwords that came into English" (as Kemmer puts it), such as anchor and butter (from the Germanic period).
Since both the words you mention (cafe and touché) originated from the French but are used by non-French people, they would be considered loanwords in English, and hence English words in their own right.
Whether the forms without the diacritical marks are misspellings is debatable. ODO lists cafe and touché in those forms, diacritical marks absent from one and present in the other. Hence cafe is definitely not a misspelling. As for touche, it is arguably a misspelling; but since English doesn't generally use diacritical marks, it may be acceptable to some. As for statistics, touche is several times more prevalent touché based on the database used by Ngram.