Reading this informational blurb from the INIS, the first step will be to register with the local Garda station.
http://www.inis.gov.ie/en/INIS/Pages/WP07000279
What you're hoping to get is a Stamp 3 for a limited amount of time (the right to reside in Ireland without permission to work or to receive state funds).
Since the Guards tend to make things up as they go along and often don't know of what they speak, I'd be prepared with the following: (1) A formal letter of sponsorship from your sister and her spouse outlining the reasons for your stay and how and why they intend to sponsor your remaining in Ireland (2) Evidence that both you and they have sufficient funds to cover your stay in Ireland (3) private health insurance for the duration of your stay (4) Evidence that you fully intend on leaving Ireland at the end of your extended stay, e.g. job offer, family commitments, financial commitments etc.
These are things I'm just writing on the spot. Don't be surprised if the Guards either refer you back to making a written application to the INIS--unfortunately, I have no idea how long that process would take.
That said, the Guard might be having a wonderful day and stamp you through without a problem. But your first point of call will be the local Garda station. They will be best suited to advise you what documents you need and how/if the process can be completed.
Best wishes.