How: Shannon knew in advance she would be killing Franziska and acquired a silenced gun to that end. However, the method didn't make itself apparent until Franziska announced she would be going through the documents in a room guarded by Kyrie. This is why it bears so many signs of improvisation (see below). As Bernkastel lived in the house, Shannon had a good understanding of Kyrie's psychology. Furthermore, as Erika had a condition that caused her pain, it is likely she had sedatives on hand to help her sleep. Shannon drugged Kyrie when delivering her tea with Kanon. Afterwards, when she split up from Kanon to "go to the kitchen," she walked past the sleeping girl and killed Franziska.
Why: As stated above, Shannon was Erika's daughter, but only recently learned, in light of the need for a bone marrow transplant, that they were related. When it became clear that Franziska, not her, was likely to be taken back in as a daughter, while Shannon would remain a maid, she snapped.
Additional notes: It is possible, though by no means certain, that Shannon's congenital condition is being unable to speak. We know that it is nothing visible to casual inspection, at least.
It is plausible that Shannon never really planned to get away with this crime in the long-term. She drugged Kyrie and used a silenced gun to reduce the chance that anyone besides her target would die. However, sooner or later, Kyrie would be bound to figure it out, and would be likely to tell someone insofar as she cared for Franziska and her fear of her mother, while strong, was not invincible. And that's assuming nobody tested Kyrie for drugs! Or found her powder-marked gloves, or worked out how she got the gun.