I couldn't access the internet connection for the past five days. So, kindly address the issue and solve it.
I haven't been accessing the internet connection for the past five days. So, kindly address the issue and solve it.
Please check them.
The problem is not the choice between "couldn't" and "haven't been". The problem is that the second sentence is vague and does not obviously connect with the first. Who are you saying this to, and why?
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The problem is not the choice between "couldn't" and "haven't been". The problem is that the second sentence is vague and does not obviously connect with the first. Who are you saying this to, and why? Use as many words as you need to explain the situation.
Try:
My internet service has been down for the past five days. Please fix it!