You've made a mistake on her birth year, I think.
I'd say you're correct on the spelling ie Mulrennan - spelling changed so much in the 19th century.
Griffiths Valuation has only one James Mulrennan in Sligo who is probably her father. However, records for that parish don't go back far enough to check her birth or parents marriage.
You could check with the Sligo Genealogy centre to see if they have any more records.
Also, does her husband have a Sligo connection too? There are Colemans in Sligo and many people emigrated and stayed among their own whilst living abroad. His history might provide some clues.