What is the difference between 〜と いいました and 〜と いって いました ?Feel free to just provide example sentences.

~といいました roughly equals "I/he/she/etc said", and ~といっていました is sometimes understood as "was saying/has or had been saying" depending on the context.

As for the examples, well, let's see if I can come up with something useful.

彼は魚が嫌いだといいました。- He said he hated fish (just stated a plain fact. He might have mentioned it only once).

彼は魚が嫌いといっていました、が彼女は魚を作りました。- He had been saying he hated fish (like, he'd always said that to his friends and was totally open about it, and kindly reminded them about it whenever they were going to eat out), but she still cooked fish (maybe she wasn't paying attention, or didn't really care).

Hope it helps a bit.
Anyway, this is as far as my knowledge goes.