The correct option is C had known
Options A and C:
The words 'even before I _____ her for a week' refer to a past perfect action. 'For a week' tells a period of time; when we use simple past tense 'knew' we describe an action that was completed in the past. We don't use the simple past in terms of time period. Hence option A is correct.
'Had known' describes a past perfect action, this tense describes an action that completed just before another commenced 'Ramya (started) asking for books even before'. Hence this verb form is suitable here and option C is correct.
Option B: 'Was known' is in the passive voice; it describes an action received by a subject.
But, the given sentence is in active voice (the word 'by' is not present here 'was known by'; this word is used in the passive voice) where the subject I performs an action.
Option D: Similarly, 'had been known' is also in the passive voice.
Hence options B and D are incorrect.