So far, in 8,000 miles of towing, I haven't had a flat, but there's always the first time. I'm not sure the jack in my truck would go under the axle of the trailer. What do you carry?
I use a jack called Good Sam Roadside Assistance. It has worked very well for me and is very easy to store, needs no lubrication, and takes up very little space. Trailer Aide works very well, but takes a lot of space.