Actually according to my Greek girlfriend this sentence is a joke to most native speakers - by saying κανω instead of φτιάχνω you are sayin:
"I'm pretending to be a cake" or "I am a cake"
Also the literal English makes it sound like "I'm having sex with a cake." So if you're "doing" food you're Jim from American Pie.
The diaeresis/tréma is not an accent mark. But in Modern Greek every word with more than one syllable must be written with an accent mark (the acute accent ´ to be precise) on the accented syllable - so the spelling *κεϊκ is not acceptable. (This is also the reason why words with a diaeresis on the accented syllable also take an accent: e.g. in the word πρωτεΐνη.)
But the spelling κέικ (with the accent on the ε!) does both: it indicates that the vowels ε and ι are not to be read as a digraph in one syllable, but as separate vowels in two syllables—which makes the diaeresis unecessary—and also provides the accent mark for the accented syllable. (Now if this word was accented on the second syllable, then as stated above we would have to spell it *κεΐκ - i.e. with both diaeresis and accent.)
Maybe cake still is πίτα in traditional Greek cakes or compound words . My Ελληνική κουζίνα has two cakes: Κέικ με κεράσια/ cake with cherries and Γιαουρτόπιτα, Σπεσιαλιτέ Μετσόβου. The last a quite ordinary cake: 2 cups with sugar, 2 with flour, 4 eggs, 300 gr yoghurt, 250 gr. butter, baking powder?
There is also the word τούρτα (I think it is specifically used to designate a cake with layers, frosting and decorations, etc. for birthdays and other special occasions). Even so, I am pretty sure τούρτα must ultimately be of Latin origin because some variation of it is the word (or one of the words) for "cake/pie" in the Romance languages (e.g. Spanish/Italian/Portuguese= "torta"; French= "tourte"; Romanian= "tort").
I agree that the word "τούρτα" should be used, instead of κέικ. Although κέικ is often used in Greece, it's simply the transliterated English word "cake". And yes, I realize that "τούρτα" is also a borrowed Latin word, (the Greek language, because of its long history and influences from neighboring countries and conquerors, borrows many words from various languages), but many Greek linguists frown upon transliteration such as this.