Believe it or not, no mirror. Practice in the safety of your own home until you can get it down. I've had contact lenses fall out in the most inconvenient of places, so out of necessity I've learned to put them back in without a mirror. Otherwise, if you keep a small signal mirror in your kit, use that.
I'm sure you already do this, but one of the tricks for making the lenses 'stick' to your eyes, so that there's a greater chance of getting it in 1 try, is to drop 1-2 drops of saline solution directly on the lens before placing them in your eye. I also, drop saline into each eye before placing the lenses in. I find that the combination of these 2 things not only makes the lens stick easier, but also provides for lasting comfort throughout the day ( no dryness ).
In addition to different brands of contacts, I've noticed that different brands of Saline Solution provide different levels of comfort. Opti-Free is like the devil to me...I swear my eyeballs can actually feel the lens when I use this brand of solution. I exclusively use Aquify. It's a bit pricier but its worth the long lasting comfort