I think your wife needs to apply for an Irish visa. https://www.dfa.ie/irish-embassy/gre...s-for-ireland/
You may as well get a passport for your little one as it will be handy just in case. Someone told me in the past that when they went to Ireland, their passports or IDs weren't checked. Another one told me they were checked. So to avoid the risk, you better have her passport in hand.