Do a trial run on a Sunday morning to see if you are happy with the route.
If you can ride 6miles without a problem, then you can extend it more. You dont have to ride hard or fast on a commute.
Your bike is plenty good enough for the journey.
I would agree that whatever your bike, high quality tyres will make your ride faster and more efficient, as well as reducing punctures. You need some kevlar banded, medium width slick tyres, kept at a good high pressure.

For a journey of that length I would recomend using cycling shorts, as well as helmet and gloves. Carry your stuff on the bike, not your back, and carry enough water. Apart from that, its only a commute, its not rocket science.

Good luck
Michael.