New sister. Now what?

I'll admit all this is fresh and new to me, so there's the potential that I'm not 100% thinking clearly. In his 20s, my father apparently had a brief relationship with someone and a child was produced. This was sometime in the early 1960s. According to what we've been told so far through an intermediary, he was likely never aware of the child's existence. I do not yet know the reason why. Approximately 8 years after the child was born, my dad and mother met. They later married and had a few children. We had a pretty average childhood and my dad was always my biggest fan/cheering team. The man I knew had a strong sense of family and a very strong sense of right, wrong and responsibility. Granted, he did become my dad quite a bit later in life, and his belief system likely would have continued to develop between the time of the prior relationship and when he married my mum. We learned of the connection through a genealogy site's DNA database. Through the intermediary, we have learned that she was adopted by members of her birth mother's family, has had some contact with her birth mother throughout her life, and has known our father's name since she was a teen (about when I was 2 years old). Since she says it's highly unlikely my father ever knew of her birth, I am left to assume she never attempted to track down her birth father prior to now. She would be in her mid-50s. I am in my early 40s. He passed away years ago, so no one can tell him of the child's existence. There's no affair or sense of betrayal when it comes to my parent's relationship. And dad could hardly have been expected to tell us or our mother if he himself didn't know. I am absolutely and utterly confused as to what to do next. She initiated contact with the intermediary, so far requesting medical history and photos. I'm told she's somewhat insistent and impatient for quick responses. After 30+ years of marriage, my mother is close to my dad's family, so there's no way to avoid telling her if my new sister wants to connect with his side of the family (or his children, nieces and nephews). However, my mother is older, not always in best health and lives 4,500 km away. Considering the emotional roller-coaster I have been on for the past 48 hours and my mother's health, this is news that I strongly feel that should be delivered in person. I can't afford to take time off work and pay thousands for a plane ticket to see her immediately, which likely means there's no way to deliver the news in person within the next 7-10 months (our next scheduled visit). I am a ball of emotions. I feel guilty for all of them. I am grateful that the birth mother's family never told my dad, otherwise he never would have met my mother years later and I wouldn't have existed. I am angry that they took the decision for him to be involved in her life away from him. I am relieved that my parents never had to deal with this situation in their marriage. I am angry that she is pressuring the intermediary for information to be delivered quickly after having spent 40+ years knowing his name and not seeking her family out until now. I feel pressured that I have to work on her timeline, with no time allowed to come to terms with this new reality. Has anyone gone through something like this? How did you move forward?