。
I heard about Keiko's problem through the grapevine.
Literal
Keiko-[honorific-さん] [attributive-の] problem [topic-は] rumor [by-で] heard-and-learned-[past].
聞き知る ('to come to know by hearing') is a compound verb — 聞く ('hear') + 知る ('know') = 'to learn about through hearing.' Used for passive acquisition of information. うわさで ('through rumor') marks the channel of information. A slightly distanced way to reference hearing something through the grapevine.