Mary Maloney is so insistent on staying in the living room because it means that she can stay in the thick of the action and find out how the search for both the murder weapon and the murderer is going. She pretends to feel unwell and uses this as a reason for staying in her chair, but really this is a very clever and well planned strategy to enable her to overhear the suspicions and the comments of the detectives as they work around her. Note how the text demonstrates this in the following quote:

So they left her there while they went about their business, searching the house. Occasionally on of the detectives asked her another question. Sometimes Jack Noonan spoke at her gently as he passed by...

This enables her to hear their whispers, but also to monitor their search for the murder weapon. She knows that if she can hide the murder weapon for ever, then it is extremely likely that she will never be convicted. Thus, until the murder weapon is consumed, she needs to stay on the scene. This means when the lamb is cooked, she is ready to encourage the detectives to eat it and ensure her safety.