“I have been sick for a week.”
This is a Simple Sentence, which means it has only one independent clause containing a single subject and a single verb. The verb phrase “have been” is Present Perfect with the helping verb [have] and main verb [been] together acting as a single unit: a verb. It is a linking verb.
Because the Adverb Prepositional Phrase “for a week” answers “How long?” it is acting as an adverb, of which adverbs modify adjectives, verbs, and other adverbs. So, in your sentence “for a week” is modifying, and can only modify, the verb phrase “have been.” The reason for this is that the verb phrase "have been" is in tense, giving reference to time, and the adjective "sick" does not have anything to do with time.
I / have been / sick / for a week
Subject / Present Perfect verb phrase / predicate adjective / adverb prepositional phrase
Adverbs, and adverbial phrases are pretty versatile and can be found in different places in a sentence, AND still modify the word it is intended to modify:
For a week, I have been sick. OR I have been, for a week, sick. [the commas are used for clarity and not to offset the phrase from the sentence because it is nonessential (the sentence can survive without it and still be meaningful: I have been sick.)]
John E. Warriner. Warriner’s English Grammar and Composition. Third Course. Liberty Edition. Orlando, Florida: Harcourt, Brace, and Jovanovich. 1986.