Native speakers would not say "I have got (UK) / gotten (US) sick for + duration."
You get or become sick at a particular time or gradually, over some time.
e.g. I got sick last Tuesday.
e.g. I have been (gradually) getting sick for the last few days (and this process may continue)
When you think that the process of "getting sick" has reached a certain stage, you might then say "Now, I am sick. I have been sick for one week / since last Saturday".
So in summary, the real-life options are:
a) "I have been sick for x days" (I am already sick and this is the duration of my sickness so far) - Most common.
b) "I have been getting sick for x days" (and this process of becoming sick may continue into the future) - Unusual but correct and would sound OK in specific contexts.