Hello.
I went to Ecuador for 4 months whilst there I travelled about a bit, and as a type 1 with all of my trips abroad I had the worry about how to keep my Insulin cool. I bought a couple of Frio Insulin Travel Wallets. No need for anything other than cold water, submerge the pack in cold water for a few minutes, then load your insulin back up in the wallet and it keeps it cool for a few days. These were so good, all my insulin was kept cool. The only advise I would give would be buy a couple of smaller ones as they keep things cool for longer.....plus it seperates your insulin, and since your going with someone else give them soon to carry just in case your baggage gets lost.
Quito was amazing! down town is especially fun I also travelled in the amazon for a while and loved every minute of it! I hope you have a fantastic time and although it sounds aweful try the Guinea Pig its a Ecuador delicacy and is really good!
Freya