Annie,
Well if Nancy was Church of Ireland and from Coleraine, you might find her baptism (and that of any siblings) in the church records. Baptisms & marriages for Coleraine COI start in 1769. I don’t think they are on-line anywhere but there is a copy in PRONI in Belfast.
Kane (and O’Kane) is a very common name in Co. Londonderry. The O’Cahans – as it was originally spelled - were the main tribe that occupied that area and there are thousands of them. (The 1831 census has 1219 in the county).
I had a look at the Valuation Revision records (a bit like the Scottish Valuation records) which are on the PRONI site. I looked at North Ramparts, Coleraine for 1874 to 1884 but didn’t see a Kane household. However several of the properties were marked “lodgers” so Nancy may well have been staying there.
I looked for a Rodger/Roderick Kane in the 1831 census. None in Coleraine. There were 2 others in the county. Both RC, and in Drumachose and Tamlaght Finlaggan, parishes some distance away from Coleraine.
39 Kanes & O'Kanes in Coleraine in the 1901 census.
I noticed that a Roderick Kane married Margaret Wilson on 12.12.1864 in Coleraine. On the basis that families often repeat names, you might want to look at that to see if there’s any connection to your family. That couple had a daughter on 26th July 1866. Father (Roddy Kane) was a labourer and the address was Killowen, so I think that might be your family.
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_retur…