What is the classical music played by Woolsey in his room at the end of the programme?
It's indeed a piece of music made for the show, and as far as I know, it doesn't have a name.
On his blog, Joseph Mallozzi, writer and producer of Stargate Atlantis (among others), wrote about "Broken Ties" just after the episode aired, and revealed that Joel Goldsmith composed that piece instead of picking up an existing one.
THE FINAL MONTAGE: Teyla’s family/Bathtub McKay/Sheppard and Ronon.
The third and final montage sequence of the episode. I gave Joel Goldsmith the choice of either finding an appropriate classical piece for this sequence or scoring it himself. To no one’s surprise, he composed a beautiful piece to compliment the montage.