I think that Kurt would probably have told his father. Klaine's break-up took place before Christmas. Kurt wasn't hiding it or his feelings about it.
My feeling during the last Christmas special (when Burt and Klaine came to NY) was that Burt was trying to bring Kurt and Blaine back together. Perhaps because of his own health scare reminding him of how precious time is. But, thinking about it now, it does seem like there's a part of the story missing with Burt, because he typically gets very protective and angry when Kurt is hurt in any way.
I suppose Blaine could have become close enough to Burt to support him through the cancer diagnosis without telling him about the cheating. But only, I imagine, if there was no piano nearby. There's just something about pianos that brings out Blaine's secrets. Once he touched those ivories, he would have collapsed in a blubbering mess and confessed everything.
About the New New Directions, only the seniors really know Kurt at all. So, that means Artie is the only friend potentially out of the loop in the choir. The others might be unhappy with Blaine in theory, but they wouldn't have the personal reaction that someone like Rachel or Mercedes might.