What is the shortest layover in Istanbul that would let me visit The Blue Mosque?

I am planning a trip that will involve changing planes in Istanbul (IST) airport. This is a routing that a travel agent chose to save my client money, and I've agreed to it because I've never been there before, and even if I don't leave the airport, it should be cool.

What would be even cooler is if I could slip out and see something - and the number one something everyone says to see is The Blue Mosque. In order to evaluate whether a given layover would work for such a purpose, I need to know:

• the minimum connecting time (coming from North America and headed into Europe, or vice versa). There will be a terminal change apparently.
• how long it takes to get to The Blue Mosque by train or taxi or whatever else is quick
• how long the lineups typically are to enter it
• how long to spend inside to feel that you at least saw it. I'm guessing 30 minutes for this though it would be nice to have longer.

Add all those up and I can evaluate whether any itinerary being proposed to me would be able to include such a side trip or not.