Native English speaker from USA.
Present perfect starts at some point in the past and goes to the present. You started getting sick sometime last week and you are still sick now. You would even say I have had (sometime starting in the past up until this moment) pneumonia three times.
If you are no longer sick right now you would use the past perfect and say I had been sick last week. Or more commonly you would just use the past and say I was sick last week.