The given famine name Ida get out from the Hebrew Biblical name Jehudith (Yudith) יהודית.
The name Ida was used in the German countries in the Middle Ages,
From them it go over to the Jews that live there.
You have a lot of variation of this name ,like Ide,Yte,Yute, and more.
The name Ida have nothing to do with the name Chaike.
Not every combination of names is the same variant of the first of second name.